Question

The adjacent figure HOPE is a parallelogram. Find the angle measures x,y,z. State the properties you use to find them.
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Answer 1

Step-by-step explanation:

As per the provided information in the given question, we have :

• HOPE is a || gm.
• Measure of ∠EHP is 40°.
• Exterior angle of parallelogram is 70°.

Let us say that ∠HOP is ∠1 and ∠EPH is ∠2.

Here, the exterior angle of the parallelogram and ∠1 is forming a linear pair. So,

$\longrightarrow \sf{\quad { \angle 1 + Exterior \; \angle = 180^\circ}}$

• Reason : Linear Pair

Substitute the value of the measure of the exterior angle.

$\longrightarrow \sf{\quad { \angle 1 + 70^\circ = 180^\circ}}$

Transposing 70° from LHS to RHS.

$\longrightarrow \sf{\quad { \angle 1 = 180^\circ - 70^\circ}}$

Performing subtraction.

$\longrightarrow \sf{\quad { \angle 1 = 110^\circ}}$

Now, we got the value of ∠1 that is 110°. Now,

$\longrightarrow \sf{\quad {x = \angle 1 }}$

• Reason : Opposite angles of a parallelogram are equal.

$\longrightarrow \quad\underline{\boxed { \textbf{\textsf{x = 110}}^\circ }}$

Now, let's find the value of z. Here,

$\longrightarrow \sf{\quad { \angle 2 = z}}$

• Reason : Alternate interior angles are equal.

So, we need to calculate the value of ∠2 first.

In ∆EHP :

$\longrightarrow \sf{\quad { 40^\circ + x + \angle 2 = 180^\circ}}$

• Reason : Sum of all the interior angles of a triangle is 180°.

Substitute the value of x.

$\longrightarrow \sf{\quad { 40^\circ + 110^\circ + \angle 2 = 180^\circ}}$

Performing addition.

$\longrightarrow \sf{\quad { 150^\circ + \angle 2 = 180^\circ}}$

Transposing the terms.

$\longrightarrow \sf{\quad { \angle 2 = 180^\circ - 150^\circ}}$

Performing subtraction.

$\longrightarrow \sf{\quad { \angle 2 = 30^\circ}}$

Now, as

$\longrightarrow \sf{\quad {z = \angle 2 }}$

Substitute the value of angle 2.

$\longrightarrow \quad\underline{\boxed { \textbf{\textsf{z = 30}}^\circ }}$

Now, finding the value of y :

$\longrightarrow \quad\underline{\boxed { \textbf{\textsf{y = 40}}^\circ }}$

• Reason : Alternate interior angles are equal.

$\underline{ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$

Therefore,

⠀⠀⠀⠀⠀★ x = 110°

⠀⠀⠀⠀⠀★ y = 40°

⠀⠀⠀⠀⠀★ z = 30°