Question
In the given figure, $AB \| CD \| EF , \angle D B G=x, \angle E D H=y, \angle A E B=z, \angle E A B=90^{\circ}$ and $\angle B E F=65^{\circ}$. Find the values of $x , y$ and $z$ .
✓
Answer
$EF \| CD$ and $ED$ is the transversal.
$\therefore \angle F E D+\angle E D H=180^{\circ} \ [$co $-$ interior angles$]$
$\Rightarrow 65^{\circ}+ y =180^{\circ}$
$\Rightarrow y =\left(180^{\circ}-65^{\circ}\right)=115^{\circ}$
Now $CH \|AG$ and $DB$ is the transversal
$\therefore x = y =115^{\circ} \ [$corresponding angles$]$
Now $,\text{ABG}$ is a straight line.
$\therefore \angle A B E+\angle E B G=180^{\circ} \ [$sum of linear pair of angles is $180^{\circ} ]$
$\Rightarrow \angle A B E+x=180^{\circ}$
We know that the sum of the angles of a triangle is $180^{\circ}$.
From $\triangle E A B$, we get
$\angle E A B+\angle A B E+\angle B E A=180^{\circ}$
$\Rightarrow 90^{\circ}+65^{\circ}+ z =180^{\circ}$
$\Rightarrow z =\left(180^{\circ}-155^{\circ}\right)=25^{\circ}$
$\therefore x =115^{\circ}, y =115^{\circ}$ and $ z =25^{\circ}$
$\therefore \angle F E D+\angle E D H=180^{\circ} \ [$co $-$ interior angles$]$
$\Rightarrow 65^{\circ}+ y =180^{\circ}$
$\Rightarrow y =\left(180^{\circ}-65^{\circ}\right)=115^{\circ}$
Now $CH \|AG$ and $DB$ is the transversal
$\therefore x = y =115^{\circ} \ [$corresponding angles$]$
Now $,\text{ABG}$ is a straight line.
$\therefore \angle A B E+\angle E B G=180^{\circ} \ [$sum of linear pair of angles is $180^{\circ} ]$
$\Rightarrow \angle A B E+x=180^{\circ}$
We know that the sum of the angles of a triangle is $180^{\circ}$.
From $\triangle E A B$, we get
$\angle E A B+\angle A B E+\angle B E A=180^{\circ}$
$\Rightarrow 90^{\circ}+65^{\circ}+ z =180^{\circ}$
$\Rightarrow z =\left(180^{\circ}-155^{\circ}\right)=25^{\circ}$
$\therefore x =115^{\circ}, y =115^{\circ}$ and $ z =25^{\circ}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $p(x) = x^3 - 3x^2 + 2x$, find $p(0), p(1), p(2)$. What do you conclude?
→Find the volume, curved surface area and the total surface area of a cone having base radius $35\ cm$ and height is $12\ cm$. $\Big(\text{Use}\ \pi=\frac{22}{7}\Big).$
→In the given figure, $AB\ ||\ CD$. Find the value of $x$.
→The number $52, 53, 54, 54, (2x + 1), 55, 55, 56, 57$ have been arranged in an ascending order and their median is $55$. Find the value of $x$ and hence find the mode of the given data.
→In the given figure, $O$ is the centre of the given circle and measure of arc $ABC$ is $100^\circ$. Determine $\angle\text{ADC}$ and $\angle\text{ABC}.$
→In the figure, $AB || PQ$. Find the value of $x$ and$ y.$
→Following table gives the distribution of students of sections $A$ and $B$ of a class according to the marks obtained by them.
Represent the marks of the students of both the sections on the same graph by two frequency polygons.What do you observe?
→|
Section A
|
Section B
|
||||||||||||||||||||||||||
|
Marks
|
Frequency
|
Marks
|
Frequency
|
||||||||||||||||||||||||
|
|
|
|
||||||||||||||||||||||||
Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.
In each of the figures given below, $ABD$ is a rectangle. Find the values of $x$ and $y$ in each case.
→Simplify: $(a + b + c)^2 + (a - b + c)^2 + (a + b - c)^2$
→