Parallel and Intersection Lines<\/strong><\/strong> \u2013 NCERT Solutions<\/h2>\n\n\n\n
\n\n\n\nQ.1: How many angles do they form?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
In Figure, where line l intersects line m, we can see that four angles are formed.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753954112-989v9h.jpg\")
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\n\n\n\nQ.2: Can two straight lines intersect at more than one point?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
No, two straight lines cannot intersect at more than one point. If two lines intersect at more than one point, then they are coincident lines.<\/p>\n\n\n\n
\n\n\n\nQ.3: What patterns do you observe among these angles?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
It is observed that the sum of adjacent angles formed by the intersection of two lines is 180o<\/sup>, and the vertically opposite angles are equal in measure.<\/p>\n\n\n\n
\n\n\n\nQ.4: In figure, if {tex}\\angle {a}{\/tex} is {tex}120^{\\circ}{\/tex}, can you figure out the measurements of {tex}\\angle b, \\angle c{\/tex} and {tex}\\angle d{\/tex}, without drawing and measuring them?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753954755-5dmgze.jpg\")
<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
We know that {tex}\\angle a{\/tex} and {tex}\\angle b{\/tex} together measure {tex}180^{\\circ}{\/tex}, because when they are combined, they form a straight angle which measures {tex}180^{\\circ}{\/tex}. So, if {tex}\\angle a{\/tex} is {tex}120^{\\circ}{\/tex}, then {tex}\\angle b{\/tex} must be {tex}60^{\\circ}{\/tex}.
Similarly, {tex}\\angle b{\/tex} and {tex}\\angle c{\/tex} together measure {tex}180^{\\circ}{\/tex}. So, if {tex}\\angle b{\/tex} is {tex}60^{\\circ}{\/tex}, then {tex}\\angle c{\/tex} must be {tex}120^{\\circ}{\/tex}. And {tex}\\angle c{\/tex} and {tex}\\angle d{\/tex} together measure {tex}180^{\\circ}{\/tex}. So, if {tex}\\angle c{\/tex} is {tex}120^{\\circ}{\/tex}, then {tex}\\angle d{\/tex} must be {tex}60^{\\circ}{\/tex}.
Therefore, in Fig. 5.2, {tex}\\angle a{\/tex} and {tex}\\angle c{\/tex} measure {tex}120^{\\circ}{\/tex}, and {tex}\\angle b{\/tex} and {tex}\\angle d{\/tex} measure {tex}60^{\\circ}{\/tex}.
When two lines intersect each other and form four angles, labelled {tex}{a}, {b}, {c}{\/tex} and d , as in Fig. 5.2, then {tex}\\angle a{\/tex} and {tex}\\angle c{\/tex} are equal, and {tex}\\angle b{\/tex} and {tex}\\angle d{\/tex} are equal!<\/p>\n\n\n\n
\n\n\n\nQ.5: Is this always true for any pair of intersecting lines?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Yes, any pair of intersecting lines forms vertically opposite angles, which are equal in measure.<\/p>\n\n\n\n
\n\n\n\nQ.6: Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753955453-uc93ny.jpg\")
If two lines intersect and all four angles are equal, then each angle must be a right angle (90o<\/sup>). Perpendicular lines are a pair of lines which intersect each other at right angles (90o<\/sup>). In Figure, we can say that lines l and m are perpendicular to each other.<\/p>\n\n\n\n
\n\n\n\nQ.7: Which pairs of lines appear to be parallel in figure below?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753955671-phm8zf.jpg\")
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Solution:<\/p>\n\n\n\n
Lines a, I, and h are parallel to each other.
Line b is parallel to line e.
Line c is parallel to line g.
Line d is parallel to line f.<\/p>\n\n\n\n
\n\n\n\nQ.8: List all the linear pairs and vertically opposite angles you observe in figure.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753955189-ndaeh7.jpg\")
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753955218-xnu5cx.jpg\")
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Solution:<\/p>\n\n\n\n
Linear pair angles: {tex}\\angle {a}{\/tex} and {tex}\\angle {b} ; \\angle {b}{\/tex} and {tex}\\angle {c} ; \\angle {c}{\/tex} and {tex}\\angle {d} ; \\angle {a}{\/tex} and {tex}\\angle {d}{\/tex}.
Vertically opposite angles: {tex}\\angle {a}{\/tex} and {tex}\\angle {c} ; \\angle {b}{\/tex} and {tex}\\angle {d}{\/tex}.<\/p>\n\n\n\n
\n\n\n\nQ.9: Draw some lines perpendicular to the lines given on the dot paper in the figure.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753956772-c9td4a.jpg\")
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Solution:<\/p>\n\n\n\n
Do it yourself.<\/p>\n\n\n\n
\n\n\n\nQ.10: In Figure, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.<\/p>\n\n\n\n
\n- How did you spot the perpendicular lines?<\/li>\n\n\n\n
- How did you spot the parallel lines?<\/li>\n<\/ol>\n\n\n\n
![\"\"\/](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753957971-n6u77z.jpg\")
<\/figure>\n\n\n\nSolution:<\/p>\n\n\n\n![\"\"\/](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753958187-xpzf33.jpg\")
<\/figure>\n\n\n\n\n- Lines that intersect at a 90o<\/sup> angle are perpendicular.<\/li>\n\n\n\n
- Lines that do not meet, no matter how far they are extended, are parallel.<\/li>\n<\/ol>\n\n\n\n
\n\n\n\nQ.11: In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Do it yourself.<\/p>\n\n\n\n
\n\n\n\nQ.12:<\/p>\n\n\n\n
Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753958568-t63h42.jpg\")
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\n- Did you find it challenging to draw some of them?<\/li>\n\n\n\n
- Which ones?<\/li>\n\n\n\n
- How did you do it?<\/li>\n<\/ol>\n\n\n\n
Solution:<\/p>\n\n\n\n![\"\"\/](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753958616-bp4n9y.jpg\")
<\/figure>\n\n\n\n\n- Yes, some line segments are a little more difficult to draw than others.<\/li>\n\n\n\n
- Line segments e, f, h, and g.<\/li>\n\n\n\n
- Parallel lines are drawn by keeping them equidistant from the given lines.<\/li>\n<\/ol>\n\n\n\n
\n\n\n\nQ.13: In Figure, which line is parallel to line a-line b or line c? How do you decide this?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753958985-p3yvd3.jpg\")
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Solution:<\/p>\n\n\n\n
Line a is parallel to line c because both lines remain equidistant from each other and do not intersect, no matter how far they are extended.<\/p>\n\n\n\n
\n\n\n\nQ.14: Can you draw a line parallel to l, that goes through point A? How will you do it with the tools from your geometry box? Describe your method.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753960224-xm52yg.jpg\")
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Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753960339-f2emga.jpg\")
Steps of construction:<\/p>\n\n\n\n
\n- Align one edge of the set square with line l.<\/li>\n\n\n\n
- Place a ruler along the other perpendicular edge.<\/li>\n\n\n\n
- Slide the set square along the ruler until it reaches point A.<\/li>\n\n\n\n
- Draw a line through A along the set square\u2019s edge.<\/li>\n\n\n\n
- This is the required line parallel to l.<\/li>\n<\/ol>\n\n\n\n
\n\n\n\nQ.15: In Figure, parallel lines l and m are intersected by the transversal t. If {tex}\\angle 6{\/tex} is {tex}135^{\\circ}{\/tex}, what are the measures of the other angles?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753960879-hv8a5f.jpg\")
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Solution:<\/p>\n\n\n\n
{tex}\\angle 6{\/tex} is {tex}135^{\\circ}{\/tex}, so {tex}\\angle 2{\/tex} is also {tex}135^{\\circ}{\/tex}, because it is the corresponding angle of {tex}\\angle 6{\/tex} and the lines {tex}l{\/tex} and m are parallel.
{tex}\\angle 8{\/tex} is {tex}135^{\\circ}{\/tex}, because it is the vertically opposite angle of {tex}\\angle 6 . {\/tex} {tex}\\angle 4{\/tex} is {tex}135^{\\circ}{\/tex} because it is the corresponding angle of {tex}\\angle 8{\/tex}.
{tex}\\angle 2{\/tex} is {tex}135^{\\circ}{\/tex} because is the vertically opposite angle of {tex}\\angle 4{\/tex}. So, {tex}\\angle 2, \\angle 4{\/tex}, {tex}\\angle 6{\/tex}, and {tex}\\angle 8{\/tex} are all {tex}135^{\\circ}{\/tex}.
{tex}\\angle 5{\/tex} and {tex}\\angle 6{\/tex} are a linear pair, together they measure {tex}180^{\\circ}{\/tex}. If {tex}\\angle 6{\/tex} is {tex}135^{\\circ}{\/tex}, then
{tex} \\angle 5=180-135=45^{\\circ} {\/tex}
We can similarly find out that {tex}\\angle 1, \\angle 3{\/tex}, and {tex}\\angle 7{\/tex} measure {tex}45^{\\circ}{\/tex}.<\/p>\n\n\n\n
\n\n\n\nQ.16: In Figure, lines {tex}l{\/tex} and {tex}m{\/tex} are intersected by the transversal {tex}t{\/tex}. If {tex}\\angle a{\/tex} is {tex}120^{\\circ}{\/tex} and {tex}\\angle f{\/tex} is {tex}70^{\\circ}{\/tex}, are lines {tex}l{\/tex} and m parallel to each other?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753961212-4pg3h9.jpg\")
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Solution:<\/p>\n\n\n\n
{tex}\\angle a{\/tex} is {tex}120^{\\circ}{\/tex}, so {tex}\\angle b{\/tex} is {tex}60^{\\circ}{\/tex} because {tex}\\angle a{\/tex} and {tex}\\angle b{\/tex} form a linear pair. {tex}\\angle b{\/tex} is a corresponding angle of {tex}\\angle f{\/tex}. If {tex}l{\/tex} and {tex}m{\/tex} are parallel, {tex}\\angle b{\/tex} should be equal to {tex}\\angle f{\/tex}, however, they are not equal.
Therefore, lines land {tex}m{\/tex} are not parallel to each other as the corresponding angles formed by the transversal {tex}t{\/tex} are not equal to each other.<\/p>\n\n\n\n
\n\n\n\nQ.17: In Figure, parallel lines {tex}l{\/tex} and {tex}m{\/tex} are intersected by the transversal {tex}t{\/tex}. If {tex}\\angle 3{\/tex} is {tex}50^{\\circ}{\/tex}, what is the measure of {tex}\\angle 6{\/tex}?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753961419-psxpcd.jpg\")
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Solution:<\/p>\n\n\n\n
{tex}\\angle 3{\/tex} is {tex}50^{\\circ}{\/tex}; therefore, {tex}\\angle 2{\/tex} is {tex}130^{\\circ}{\/tex}, because {tex}\\angle 2{\/tex} and {tex}\\angle 3{\/tex} form a linear pair, and linear pairs always add up to {tex}180^{\\circ}{\/tex}.
{tex}\\angle 2{\/tex} and {tex}\\angle 6{\/tex} are corresponding angles, and they need to be equal since lines {tex}l{\/tex} and {tex}m{\/tex} are parallel.
So, {tex}\\angle 6{\/tex} is {tex}130^{\\circ}{\/tex}.
Angles {tex}\\angle 3{\/tex} and {tex}\\angle 6{\/tex} are called interior angles.<\/strong><\/p>\n\n\n\n
\n\n\n\nQ.18: In Figure, line segment {tex}A B{\/tex} is parallel to {tex}C D{\/tex} and {tex}A D{\/tex} is parallel to {tex}{BC} . \\angle {DAC}{\/tex} is {tex}65^{\\circ}{\/tex} and {tex}\\angle {ADC}{\/tex} is {tex}60^{\\circ}{\/tex}. What are the measure of angles {tex}\\angle {CAB}, \\angle {ABC}{\/tex}, and {tex}\\angle {BCD}{\/tex}?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753962250-khfaam.jpg\")
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Solution:<\/p>\n\n\n\n
Let us observe the parallel lines AB and {tex}{CD} {\/tex}. AD is a transversal of these two lines.
We know that the sum of the interior angles formed by a transversal on a pair of parallel lines adds up to {tex}180^{\\circ}{\/tex}. So
{tex} \\angle {ADC}+\\angle {DAB}=180^{\\circ} {\/tex}
{tex} 60^{\\circ}+\\angle {DAB}=180^{\\circ} . {\/tex}
So {tex}\\angle {DAB}=120^{\\circ}{\/tex}.
Can we find {tex}\\angle {CAB}{\/tex} from this?
{tex} \\angle {DAB}=\\angle {DAC}+\\angle {CAB} {\/tex}
So {tex}120^{\\circ}=65^{\\circ}+\\angle {CAB}{\/tex}.
So {tex}\\angle {CAB}=55^{\\circ}{\/tex}.
Let us observe the parallel line segments {tex}A D{\/tex} and {tex}B C{\/tex}. They are intersected by a transversal CD. So, {tex}\\angle {ADC}+\\angle {BCD}=180^{\\circ}{\/tex}, because they are interior angles on the same side of the transversal. Since {tex}\\angle {ADC}{\/tex} is given as {tex}60^{\\circ}, \\angle {BCD}=120^{\\circ}{\/tex}
Similarly, we find {tex}\\angle A B C=60^{\\circ}{\/tex}.
Therefore, in Fig. 5.29, {tex}\\angle {CAB}=55^{\\circ}, \\angle {ABC}=60^{\\circ}{\/tex}, and {tex}\\angle {BCD}=120^{\\circ}{\/tex}.<\/p>\n\n\n\n
\n\n\n\nQ.19:<\/p>\n\n\n\n
\nFind the angle marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753963360-vgm2wk.jpg\")
<\/p>\n<\/blockquote>\n\n\n\n
Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle a is 48o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.20: Find the angle\u00a0marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753963858-xr6x7w.jpg\")
<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle b is 52o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.21: Find the angles marked below
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753964115-94p6bu.jpg\")
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Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle c is 81o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.22: Find the angle\u00a0marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753964231-vdh83y.jpg\")
<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle d is 99o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.23: Find the angle\u00a0marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753964377-3fwur4.jpg\")
<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle e is 69o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.24: Find the angle marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753964740-nhcmy5.jpg\")
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Solution:<\/p>\n\n\n\n
Since the sum of interior angles on the same side of a transversal is always equal to {tex}180^{\\circ}{\/tex}. Therefore,
{tex}f{\/tex} {tex} +132^{\\circ}=180^{\\circ} {\/tex}
{tex} f=180^{\\circ}-132^{\\circ} {\/tex}
{tex} f=48^{\\circ} {\/tex}<\/p>\n\n\n\n
\n\n\n\nQ.25: Find the angle marked below
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753964783-735hpa.jpg\")
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Solution:<\/p>\n\n\n\n
Since corresponding angles formed by a transversal intersecting a pair of parallel sides are equal, angle g is 122o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.26: Find the angle marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753964909-evcrws.jpg\")
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Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle h is 75o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.27: Find the angle\u00a0marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753965004-pxfj83.jpg\")
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Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle i is 54o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.28: Find the angle\u00a0marked below.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753965088-83gssg.jpg\")
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Solution:<\/p>\n\n\n\n
Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle j is 97o<\/sup>.<\/p>\n\n\n\n
\n\n\n\nQ.29: Find the angle represented by a.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753965675-p4jmfe.jpg\")
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Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753965816-aj3d57.jpg\")
{tex}\\angle 1=\\angle 2=42^{\\circ}{\/tex} …(Vertically opposite angles)
Line p is parallel to q, and s is a transversal, then
{tex}{a}+\\angle 1=180^{\\circ}{\/tex} …(Sum of interior angles on the same side of the transversal)
{tex}{a}+42^{\\circ}=180^{\\circ}{\/tex}
{tex}{a}=180^{\\circ}-42^{\\circ}{\/tex}
{tex}a=138^{\\circ}{\/tex}<\/p>\n\n\n\n
\n\n\n\nQ.30: Find the angle represented by a.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753965942-zh5rp5.jpg\")
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Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753965974-kw5mvd.jpg\")
Line I is parallel to line {tex}m{\/tex}, and {tex}s{\/tex} is a transversal, then
{tex}\\angle 1=\\angle 2=62^{\\circ}{\/tex} …(Corresponding angles)
Line {tex}r{\/tex} is parallel to line {tex}s{\/tex}, and {tex}m{\/tex} is a transversal, then
{tex}\\angle 1=\\angle 3=62^{\\circ}{\/tex} …(Corresponding angles)
{tex}a+\\angle 3=180^{\\circ}{\/tex} …(Linear pair angles)
{tex}{a}+62^{\\circ}=180^{\\circ}{\/tex}
{tex}a=180^{\\circ}-62^{\\circ}{\/tex}
{tex}a=118^{\\circ}{\/tex}<\/p>\n\n\n\n
\n\n\n\nQ.31: Find the angle represented by a.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753966157-2662ez.jpg\")
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Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753966234-mbn854.jpg\")
{tex}\\angle 1=\\angle 2=110^{\\circ} \\ldots{\/tex}. (Vertically opposite angles)
Line x is parallel to line y and b is a transversal, then
{tex}\\angle 2=35^{\\circ}+\\angle 3{\/tex} … (Alternate interior angles)
or {tex}110^{\\circ}=35^{\\circ}+\\angle 3{\/tex}
{tex} 110^{\\circ}-35^{\\circ}=\\angle 3 {\/tex}
or {tex}\\angle 3=75^{\\circ}{\/tex}.
Line y is parallel to line z and c is transversal, then
{tex}\\angle 3=\\angle 4=75^{\\circ}{\/tex} …(Corresponding angles)
{tex}{a}+\\angle 4=180^{\\circ}{\/tex} …(Linear pair angles)
a {tex}+75^{\\circ}=180^{\\circ}{\/tex}
{tex}{a}=180^{\\circ}-75^{\\circ}{\/tex}
{tex}{a}=105^{\\circ}{\/tex}.<\/p>\n\n\n\n
\n\n\n\nQ.32: Find the angle represented by a.
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753966449-24nhjb.jpg\")
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Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1753966702-4gxyhh.jpg\")
{tex} \\angle 1+67^{\\circ}+\\angle 2=180^{\\circ} \\ldots {\/tex}(Sum of angles on a straight line)
{tex} \\angle 1+67^{\\circ}+90^{\\circ}=180^{\\circ} {\/tex}
{tex} \\angle 1+157^{\\circ}=180^{\\circ} {\/tex}
{tex} \\angle 1=180^{\\circ}-157^{\\circ} {\/tex}
{tex} \\angle 1=23^{\\circ} . {\/tex}
{tex} \\angle 1=\\angle {a}=23^{\\circ} \\ldots {\/tex}(Alternate interior angles)<\/p>\n\n\n\n
\n\n\n\nQ.33: In the figure\u00a0below, what angles do x and y stand for?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1754021002-yh3nzu.jpg\")
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Solution:<\/p>\n\n\n\n
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1754021257-3388st.jpg\")
Line m is parallel to line n and a is a transversal, then
{tex} \\angle 2=65^{\\circ}+\\angle 1 {\/tex} …(Corresponding angles)
{tex} 90^{\\circ}=65^{\\circ}+\\angle 1 {\/tex}
{tex} \\angle 1=90^{\\circ}-65^{\\circ} {\/tex}
{tex} \\angle 1=25^{\\circ} {\/tex}
{tex}\\angle 1={x}=25^{\\circ}{\/tex} …(Vertically opposite angles)
Line {tex}m{\/tex} is parallel to line {tex}n{\/tex} and {tex}b{\/tex} is a transversal, then
{tex}\\angle 1+\\angle y=180^{\\circ}{\/tex} …(Sum of interior angles on the same side of the transversal)
{tex} 25^{\\circ}+\\angle y=180^{\\circ} {\/tex}
{tex} \\angle y=180^{\\circ}-25^{\\circ} {\/tex}
{tex} \\angle y=155^{\\circ} {\/tex}<\/p>\n\n\n\n
\n\n\n\nQ.34: In the figure\u00a0below, what angles do x and y stand for?
![](\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/question_images\/1754021536-9tbteg.jpg\")
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Solution:<\/p>\n\n\n\n
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Line {tex}a{\/tex} is parallel to line {tex}b{\/tex} and {tex}d{\/tex} is a transversal, then
{tex}\\angle 2=\\angle 3=53^{\\circ}{\/tex} …(Alternate interior angles)
Also, line a is parallel to line b and c is a transversal, then
{tex}\\angle 1+\\angle 2=\\angle 4{\/tex} …(Alternate interior angles)
or {tex}\\angle 1+53^{\\circ}=78^{\\circ}{\/tex}
{tex} \\angle 1=78^{\\circ}-53^{\\circ} {\/tex}
{tex} \\angle 1=25^{\\circ} {\/tex}
Therefore, {tex}\\angle 1={x}=25^{\\circ}{\/tex} …(Vertically opposite angles).<\/p>\n\n\n\n
\n\n\n\nQ.35: In Figure, {tex}\\angle {ABC}=45^{\\circ}{\/tex} and {tex}\\angle {IKJ}=78^{\\circ}{\/tex}. Find angles {tex}\\angle {GEH}, \\angle {HEF}{\/tex}, {tex}\\angle {FED}{\/tex}.
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Solution:<\/p>\n\n\n\n
{tex} \\angle {ABC}=\\angle {KBE}=45^{\\circ} \\ldots \\text { } {\/tex}(Vertically opposite angles)
{tex} \\angle {IKJ}=\\angle {BKE}=78^{\\circ} \\text {… } {\/tex}(Vertically opposite angles)
{tex} \\angle {BKE}=\\angle {FED}=78^{\\circ} \\ldots \\text { } {\/tex}(Corresponding angles)
{tex} \\angle {KBE}=\\angle {BED}=45^{\\circ} \\text {… } {\/tex}(Alternate interior angles)
{tex} \\angle {BED}=\\angle {GEH}=45^{\\circ} \\text {… } {\/tex}(Vertically opposite angles)
{tex} \\angle {GEH}+\\angle {HEF}+\\angle {FED}=180^{\\circ} \\text {… } {\/tex}(Sum of angles on a straight line)
{tex} 45^{\\circ}+\\angle {HEF}+78^{\\circ}=180^{\\circ} {\/tex}
{tex} 123^{\\circ}+\\angle {HEF}=180^{\\circ} {\/tex}
{tex} \\angle {HEF}=180^{\\circ}-123^{\\circ} {\/tex}
{tex} \\angle {HEF}=57^{\\circ} {\/tex}<\/p>\n\n\n\n
\n\n\n\nQ.36: In Figure, {tex}A B{\/tex} is parallel to CD and CD is parallel to EF.\u00a0Also, EA is perpendicular to AB. If {tex}\\angle {BEF}=55^{\\circ}{\/tex}, find the values of {tex}x{\/tex} and {tex}y{\/tex}.
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Solution:<\/p>\n\n\n\n
Since EF is parallel to CD and BE is a transversal, then
{tex}\\angle {BEF}+{y}=180^{\\circ}{\/tex} …(Sum of interior angles on the same side of the transversal)
{tex} 55^{\\circ}+y=180^{\\circ} {\/tex}
{tex} y=180^{\\circ}-55^{\\circ} {\/tex}
{tex} y=125^{\\circ} {\/tex}
Since {tex}A B{\/tex} is parallel to {tex}C D{\/tex} and {tex}B E{\/tex} is a transversal, then
{tex}x=y=125^{\\circ}{\/tex} …(Corresponding angles).<\/p>\n\n\n\n
\n\n\n\nQ.37: What is the measure of angle {tex}\\angle {NOP}{\/tex} in Figure?
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Solution:<\/p>\n\n\n\n
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Q.4: In figure, if {tex}\\angle {a}{\/tex} is {tex}120^{\\circ}{\/tex}, can you figure out the measurements of {tex}\\angle b, \\angle c{\/tex} and {tex}\\angle d{\/tex}, without drawing and measuring them?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
We know that {tex}\\angle a{\/tex} and {tex}\\angle b{\/tex} together measure {tex}180^{\\circ}{\/tex}, because when they are combined, they form a straight angle which measures {tex}180^{\\circ}{\/tex}. So, if {tex}\\angle a{\/tex} is {tex}120^{\\circ}{\/tex}, then {tex}\\angle b{\/tex} must be {tex}60^{\\circ}{\/tex}.
Similarly, {tex}\\angle b{\/tex} and {tex}\\angle c{\/tex} together measure {tex}180^{\\circ}{\/tex}. So, if {tex}\\angle b{\/tex} is {tex}60^{\\circ}{\/tex}, then {tex}\\angle c{\/tex} must be {tex}120^{\\circ}{\/tex}. And {tex}\\angle c{\/tex} and {tex}\\angle d{\/tex} together measure {tex}180^{\\circ}{\/tex}. So, if {tex}\\angle c{\/tex} is {tex}120^{\\circ}{\/tex}, then {tex}\\angle d{\/tex} must be {tex}60^{\\circ}{\/tex}.
Therefore, in Fig. 5.2, {tex}\\angle a{\/tex} and {tex}\\angle c{\/tex} measure {tex}120^{\\circ}{\/tex}, and {tex}\\angle b{\/tex} and {tex}\\angle d{\/tex} measure {tex}60^{\\circ}{\/tex}.
When two lines intersect each other and form four angles, labelled {tex}{a}, {b}, {c}{\/tex} and d , as in Fig. 5.2, then {tex}\\angle a{\/tex} and {tex}\\angle c{\/tex} are equal, and {tex}\\angle b{\/tex} and {tex}\\angle d{\/tex} are equal!<\/p>\n\n\n\n
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Q.5: Is this always true for any pair of intersecting lines?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Yes, any pair of intersecting lines forms vertically opposite angles, which are equal in measure.<\/p>\n\n\n\n
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Q.6: Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?<\/p>\n\n\n\n
Solution:<\/p>\n\n\n\n
Q.7: Which pairs of lines appear to be parallel in figure below? Solution:<\/p>\n\n\n\n Lines a, I, and h are parallel to each other. Q.8: List all the linear pairs and vertically opposite angles you observe in figure. Solution:<\/p>\n\n\n\n Linear pair angles: {tex}\\angle {a}{\/tex} and {tex}\\angle {b} ; \\angle {b}{\/tex} and {tex}\\angle {c} ; \\angle {c}{\/tex} and {tex}\\angle {d} ; \\angle {a}{\/tex} and {tex}\\angle {d}{\/tex}. Q.9: Draw some lines perpendicular to the lines given on the dot paper in the figure. Solution:<\/p>\n\n\n\n Do it yourself.<\/p>\n\n\n\n Q.10: In Figure, mark the parallel lines using the notation given above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.<\/p>\n\n\n\n Solution:<\/p>\n\n\n\n Q.11: In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.<\/p>\n\n\n\n Solution:<\/p>\n\n\n\n Do it yourself.<\/p>\n\n\n\n Q.12:<\/p>\n\n\n\n Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper. Solution:<\/p>\n\n\n\n Q.13: In Figure, which line is parallel to line a-line b or line c? How do you decide this? Solution:<\/p>\n\n\n\n Line a is parallel to line c because both lines remain equidistant from each other and do not intersect, no matter how far they are extended.<\/p>\n\n\n\n Q.14: Can you draw a line parallel to l, that goes through point A? How will you do it with the tools from your geometry box? Describe your method. Solution:<\/p>\n\n\n\n Q.15: In Figure, parallel lines l and m are intersected by the transversal t. If {tex}\\angle 6{\/tex} is {tex}135^{\\circ}{\/tex}, what are the measures of the other angles? Solution:<\/p>\n\n\n\n {tex}\\angle 6{\/tex} is {tex}135^{\\circ}{\/tex}, so {tex}\\angle 2{\/tex} is also {tex}135^{\\circ}{\/tex}, because it is the corresponding angle of {tex}\\angle 6{\/tex} and the lines {tex}l{\/tex} and m are parallel. Q.16: In Figure, lines {tex}l{\/tex} and {tex}m{\/tex} are intersected by the transversal {tex}t{\/tex}. If {tex}\\angle a{\/tex} is {tex}120^{\\circ}{\/tex} and {tex}\\angle f{\/tex} is {tex}70^{\\circ}{\/tex}, are lines {tex}l{\/tex} and m parallel to each other? Solution:<\/p>\n\n\n\n {tex}\\angle a{\/tex} is {tex}120^{\\circ}{\/tex}, so {tex}\\angle b{\/tex} is {tex}60^{\\circ}{\/tex} because {tex}\\angle a{\/tex} and {tex}\\angle b{\/tex} form a linear pair. {tex}\\angle b{\/tex} is a corresponding angle of {tex}\\angle f{\/tex}. If {tex}l{\/tex} and {tex}m{\/tex} are parallel, {tex}\\angle b{\/tex} should be equal to {tex}\\angle f{\/tex}, however, they are not equal. Q.17: In Figure, parallel lines {tex}l{\/tex} and {tex}m{\/tex} are intersected by the transversal {tex}t{\/tex}. If {tex}\\angle 3{\/tex} is {tex}50^{\\circ}{\/tex}, what is the measure of {tex}\\angle 6{\/tex}? Solution:<\/p>\n\n\n\n {tex}\\angle 3{\/tex} is {tex}50^{\\circ}{\/tex}; therefore, {tex}\\angle 2{\/tex} is {tex}130^{\\circ}{\/tex}, because {tex}\\angle 2{\/tex} and {tex}\\angle 3{\/tex} form a linear pair, and linear pairs always add up to {tex}180^{\\circ}{\/tex}. Q.18: In Figure, line segment {tex}A B{\/tex} is parallel to {tex}C D{\/tex} and {tex}A D{\/tex} is parallel to {tex}{BC} . \\angle {DAC}{\/tex} is {tex}65^{\\circ}{\/tex} and {tex}\\angle {ADC}{\/tex} is {tex}60^{\\circ}{\/tex}. What are the measure of angles {tex}\\angle {CAB}, \\angle {ABC}{\/tex}, and {tex}\\angle {BCD}{\/tex}? Solution:<\/p>\n\n\n\n Let us observe the parallel lines AB and {tex}{CD} {\/tex}. AD is a transversal of these two lines. Q.19:<\/p>\n\n\n\n Find the angle marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle a is 48o<\/sup>.<\/p>\n\n\n\n Q.20: Find the angle\u00a0marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle b is 52o<\/sup>.<\/p>\n\n\n\n Q.21: Find the angles marked below Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle c is 81o<\/sup>.<\/p>\n\n\n\n Q.22: Find the angle\u00a0marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle d is 99o<\/sup>.<\/p>\n\n\n\n Q.23: Find the angle\u00a0marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle e is 69o<\/sup>.<\/p>\n\n\n\n Q.24: Find the angle marked below. Solution:<\/p>\n\n\n\n Since the sum of interior angles on the same side of a transversal is always equal to {tex}180^{\\circ}{\/tex}. Therefore, Q.25: Find the angle marked below Solution:<\/p>\n\n\n\n Since corresponding angles formed by a transversal intersecting a pair of parallel sides are equal, angle g is 122o<\/sup>.<\/p>\n\n\n\n Q.26: Find the angle marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle h is 75o<\/sup>.<\/p>\n\n\n\n Q.27: Find the angle\u00a0marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle i is 54o<\/sup>.<\/p>\n\n\n\n Q.28: Find the angle\u00a0marked below. Solution:<\/p>\n\n\n\n Since alternate interior angles formed by a transversal intersecting a pair of parallel lines are equal, angle j is 97o<\/sup>.<\/p>\n\n\n\n Q.29: Find the angle represented by a. Solution:<\/p>\n\n\n\n Q.30: Find the angle represented by a. Solution:<\/p>\n\n\n\n Q.31: Find the angle represented by a. Solution:<\/p>\n\n\n\n Q.32: Find the angle represented by a. Solution:<\/p>\n\n\n\n Q.33: In the figure\u00a0below, what angles do x and y stand for? Solution:<\/p>\n\n\n\n Q.34: In the figure\u00a0below, what angles do x and y stand for? Solution:<\/p>\n\n\n\n Q.35: In Figure, {tex}\\angle {ABC}=45^{\\circ}{\/tex} and {tex}\\angle {IKJ}=78^{\\circ}{\/tex}. Find angles {tex}\\angle {GEH}, \\angle {HEF}{\/tex}, {tex}\\angle {FED}{\/tex}. Solution:<\/p>\n\n\n\n {tex} \\angle {ABC}=\\angle {KBE}=45^{\\circ} \\ldots \\text { } {\/tex}(Vertically opposite angles) Q.36: In Figure, {tex}A B{\/tex} is parallel to CD and CD is parallel to EF.\u00a0Also, EA is perpendicular to AB. If {tex}\\angle {BEF}=55^{\\circ}{\/tex}, find the values of {tex}x{\/tex} and {tex}y{\/tex}. Solution:<\/p>\n\n\n\n Since EF is parallel to CD and BE is a transversal, then Q.37: What is the measure of angle {tex}\\angle {NOP}{\/tex} in Figure? Solution:<\/p>\n\n\n\n
If two lines intersect and all four angles are equal, then each angle must be a right angle (90o<\/sup>). Perpendicular lines are a pair of lines which intersect each other at right angles (90o<\/sup>). In Figure, we can say that lines l and m are perpendicular to each other.<\/p>\n\n\n\n
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Line b is parallel to line e.
Line c is parallel to line g.
Line d is parallel to line f.<\/p>\n\n\n\n
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Vertically opposite angles: {tex}\\angle {a}{\/tex} and {tex}\\angle {c} ; \\angle {b}{\/tex} and {tex}\\angle {d}{\/tex}.<\/p>\n\n\n\n
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Steps of construction:<\/p>\n\n\n\n\n
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{tex}\\angle 8{\/tex} is {tex}135^{\\circ}{\/tex}, because it is the vertically opposite angle of {tex}\\angle 6 . {\/tex} {tex}\\angle 4{\/tex} is {tex}135^{\\circ}{\/tex} because it is the corresponding angle of {tex}\\angle 8{\/tex}.
{tex}\\angle 2{\/tex} is {tex}135^{\\circ}{\/tex} because is the vertically opposite angle of {tex}\\angle 4{\/tex}. So, {tex}\\angle 2, \\angle 4{\/tex}, {tex}\\angle 6{\/tex}, and {tex}\\angle 8{\/tex} are all {tex}135^{\\circ}{\/tex}.
{tex}\\angle 5{\/tex} and {tex}\\angle 6{\/tex} are a linear pair, together they measure {tex}180^{\\circ}{\/tex}. If {tex}\\angle 6{\/tex} is {tex}135^{\\circ}{\/tex}, then
{tex} \\angle 5=180-135=45^{\\circ} {\/tex}
We can similarly find out that {tex}\\angle 1, \\angle 3{\/tex}, and {tex}\\angle 7{\/tex} measure {tex}45^{\\circ}{\/tex}.<\/p>\n\n\n\n
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Therefore, lines land {tex}m{\/tex} are not parallel to each other as the corresponding angles formed by the transversal {tex}t{\/tex} are not equal to each other.<\/p>\n\n\n\n
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{tex}\\angle 2{\/tex} and {tex}\\angle 6{\/tex} are corresponding angles, and they need to be equal since lines {tex}l{\/tex} and {tex}m{\/tex} are parallel.
So, {tex}\\angle 6{\/tex} is {tex}130^{\\circ}{\/tex}.
Angles {tex}\\angle 3{\/tex} and {tex}\\angle 6{\/tex} are called interior angles.<\/strong><\/p>\n\n\n\n
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We know that the sum of the interior angles formed by a transversal on a pair of parallel lines adds up to {tex}180^{\\circ}{\/tex}. So
{tex} \\angle {ADC}+\\angle {DAB}=180^{\\circ} {\/tex}
{tex} 60^{\\circ}+\\angle {DAB}=180^{\\circ} . {\/tex}
So {tex}\\angle {DAB}=120^{\\circ}{\/tex}.
Can we find {tex}\\angle {CAB}{\/tex} from this?
{tex} \\angle {DAB}=\\angle {DAC}+\\angle {CAB} {\/tex}
So {tex}120^{\\circ}=65^{\\circ}+\\angle {CAB}{\/tex}.
So {tex}\\angle {CAB}=55^{\\circ}{\/tex}.
Let us observe the parallel line segments {tex}A D{\/tex} and {tex}B C{\/tex}. They are intersected by a transversal CD. So, {tex}\\angle {ADC}+\\angle {BCD}=180^{\\circ}{\/tex}, because they are interior angles on the same side of the transversal. Since {tex}\\angle {ADC}{\/tex} is given as {tex}60^{\\circ}, \\angle {BCD}=120^{\\circ}{\/tex}
Similarly, we find {tex}\\angle A B C=60^{\\circ}{\/tex}.
Therefore, in Fig. 5.29, {tex}\\angle {CAB}=55^{\\circ}, \\angle {ABC}=60^{\\circ}{\/tex}, and {tex}\\angle {BCD}=120^{\\circ}{\/tex}.<\/p>\n\n\n\n
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{tex}f{\/tex} {tex} +132^{\\circ}=180^{\\circ} {\/tex}
{tex} f=180^{\\circ}-132^{\\circ} {\/tex}
{tex} f=48^{\\circ} {\/tex}<\/p>\n\n\n\n
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{tex}\\angle 1=\\angle 2=42^{\\circ}{\/tex} …(Vertically opposite angles)
Line p is parallel to q, and s is a transversal, then
{tex}{a}+\\angle 1=180^{\\circ}{\/tex} …(Sum of interior angles on the same side of the transversal)
{tex}{a}+42^{\\circ}=180^{\\circ}{\/tex}
{tex}{a}=180^{\\circ}-42^{\\circ}{\/tex}
{tex}a=138^{\\circ}{\/tex}<\/p>\n\n\n\n
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Line I is parallel to line {tex}m{\/tex}, and {tex}s{\/tex} is a transversal, then
{tex}\\angle 1=\\angle 2=62^{\\circ}{\/tex} …(Corresponding angles)
Line {tex}r{\/tex} is parallel to line {tex}s{\/tex}, and {tex}m{\/tex} is a transversal, then
{tex}\\angle 1=\\angle 3=62^{\\circ}{\/tex} …(Corresponding angles)
{tex}a+\\angle 3=180^{\\circ}{\/tex} …(Linear pair angles)
{tex}{a}+62^{\\circ}=180^{\\circ}{\/tex}
{tex}a=180^{\\circ}-62^{\\circ}{\/tex}
{tex}a=118^{\\circ}{\/tex}<\/p>\n\n\n\n
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{tex}\\angle 1=\\angle 2=110^{\\circ} \\ldots{\/tex}. (Vertically opposite angles)
Line x is parallel to line y and b is a transversal, then
{tex}\\angle 2=35^{\\circ}+\\angle 3{\/tex} … (Alternate interior angles)
or {tex}110^{\\circ}=35^{\\circ}+\\angle 3{\/tex}
{tex} 110^{\\circ}-35^{\\circ}=\\angle 3 {\/tex}
or {tex}\\angle 3=75^{\\circ}{\/tex}.
Line y is parallel to line z and c is transversal, then
{tex}\\angle 3=\\angle 4=75^{\\circ}{\/tex} …(Corresponding angles)
{tex}{a}+\\angle 4=180^{\\circ}{\/tex} …(Linear pair angles)
a {tex}+75^{\\circ}=180^{\\circ}{\/tex}
{tex}{a}=180^{\\circ}-75^{\\circ}{\/tex}
{tex}{a}=105^{\\circ}{\/tex}.<\/p>\n\n\n\n
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{tex} \\angle 1+67^{\\circ}+\\angle 2=180^{\\circ} \\ldots {\/tex}(Sum of angles on a straight line)
{tex} \\angle 1+67^{\\circ}+90^{\\circ}=180^{\\circ} {\/tex}
{tex} \\angle 1+157^{\\circ}=180^{\\circ} {\/tex}
{tex} \\angle 1=180^{\\circ}-157^{\\circ} {\/tex}
{tex} \\angle 1=23^{\\circ} . {\/tex}
{tex} \\angle 1=\\angle {a}=23^{\\circ} \\ldots {\/tex}(Alternate interior angles)<\/p>\n\n\n\n
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Line m is parallel to line n and a is a transversal, then
{tex} \\angle 2=65^{\\circ}+\\angle 1 {\/tex} …(Corresponding angles)
{tex} 90^{\\circ}=65^{\\circ}+\\angle 1 {\/tex}
{tex} \\angle 1=90^{\\circ}-65^{\\circ} {\/tex}
{tex} \\angle 1=25^{\\circ} {\/tex}
{tex}\\angle 1={x}=25^{\\circ}{\/tex} …(Vertically opposite angles)
Line {tex}m{\/tex} is parallel to line {tex}n{\/tex} and {tex}b{\/tex} is a transversal, then
{tex}\\angle 1+\\angle y=180^{\\circ}{\/tex} …(Sum of interior angles on the same side of the transversal)
{tex} 25^{\\circ}+\\angle y=180^{\\circ} {\/tex}
{tex} \\angle y=180^{\\circ}-25^{\\circ} {\/tex}
{tex} \\angle y=155^{\\circ} {\/tex}<\/p>\n\n\n\n
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Line {tex}a{\/tex} is parallel to line {tex}b{\/tex} and {tex}d{\/tex} is a transversal, then
{tex}\\angle 2=\\angle 3=53^{\\circ}{\/tex} …(Alternate interior angles)
Also, line a is parallel to line b and c is a transversal, then
{tex}\\angle 1+\\angle 2=\\angle 4{\/tex} …(Alternate interior angles)
or {tex}\\angle 1+53^{\\circ}=78^{\\circ}{\/tex}
{tex} \\angle 1=78^{\\circ}-53^{\\circ} {\/tex}
{tex} \\angle 1=25^{\\circ} {\/tex}
Therefore, {tex}\\angle 1={x}=25^{\\circ}{\/tex} …(Vertically opposite angles).<\/p>\n\n\n\n
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{tex} \\angle {IKJ}=\\angle {BKE}=78^{\\circ} \\text {… } {\/tex}(Vertically opposite angles)
{tex} \\angle {BKE}=\\angle {FED}=78^{\\circ} \\ldots \\text { } {\/tex}(Corresponding angles)
{tex} \\angle {KBE}=\\angle {BED}=45^{\\circ} \\text {… } {\/tex}(Alternate interior angles)
{tex} \\angle {BED}=\\angle {GEH}=45^{\\circ} \\text {… } {\/tex}(Vertically opposite angles)
{tex} \\angle {GEH}+\\angle {HEF}+\\angle {FED}=180^{\\circ} \\text {… } {\/tex}(Sum of angles on a straight line)
{tex} 45^{\\circ}+\\angle {HEF}+78^{\\circ}=180^{\\circ} {\/tex}
{tex} 123^{\\circ}+\\angle {HEF}=180^{\\circ} {\/tex}
{tex} \\angle {HEF}=180^{\\circ}-123^{\\circ} {\/tex}
{tex} \\angle {HEF}=57^{\\circ} {\/tex}<\/p>\n\n\n\n
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{tex}\\angle {BEF}+{y}=180^{\\circ}{\/tex} …(Sum of interior angles on the same side of the transversal)
{tex} 55^{\\circ}+y=180^{\\circ} {\/tex}
{tex} y=180^{\\circ}-55^{\\circ} {\/tex}
{tex} y=125^{\\circ} {\/tex}
Since {tex}A B{\/tex} is parallel to {tex}C D{\/tex} and {tex}B E{\/tex} is a transversal, then
{tex}x=y=125^{\\circ}{\/tex} …(Corresponding angles).<\/p>\n\n\n\n
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