Question

In the figure, AB = AC and AD || BC, find the respective measures of x, y and z.

• A. $x={70}^{\circ },y={70}^{\circ },z={55}^{\circ }$
• B. $x={70}^{\circ },y={55}^{\circ },z={55}^{\circ }$
• C. $x={55}^{\circ },y={55}^{\circ },z={55}^{\circ }$
• D. $x={70}^{\circ },y={55}^{\circ },z={70}^{\circ }$

Answer 1

The correct option is B

$x={70}^{\circ },y={55}^{\circ },z={55}^{\circ }$

Given: AD || BC and AB = AC
$△$ ABC is an isosceles triangle as its two sides are equal in length.
$\therefore$ y = z [Base angles of an isosceles triangle are equal]
In $△$ ABC,
x + y + z = 180${}^{\circ }$
x + 2y = 180${}^{\circ }$ ...(i)
$\angle EAD$ + $\angle DAB$ + $\angle BAC$ = 180${}^{\circ }$
[Angles on a straight line]
55${}^{\circ }$ + x + $\angle DAB$ = 180${}^{\circ }$
$\because$ AD || BC, $\angle DAB$ = y
$\therefore$ 55${}^{\circ }$ + x + y = 180${}^{\circ }$
x + y = 125 ...(ii)
Solving (i) and (ii)
x + 2y = 180${}^{\circ }$
2x + 2y = 250
- - -
--------------------------
-x = -70
--------------------------
$\Rightarrow x={70}^{\circ }$
$\Rightarrow y=125-x={55}^{\circ }$
$\Rightarrow z=y={55}^{\circ }$
$\therefore$ x, y and z are 70${}^{\circ }$, 55${}^{\circ }$ and 55${}^{\circ }$ respectively.