Chapter 5: Q. 19 (page 199)

For the following figure, given that $\bar{P R} | | \bar{V O}; \bar{R O} | | \bar{P V}; \bar{P R} ≅ \bar{R O}$ . Prove that $∠ 1$ and $∠ 2$ are complementary.

Short Answer

Expert verified

Therefore $∠ 1$ and $∠ 2$ are complementary.

Step by step solution

Step 1- Check the figure.

Consider the figure.

Step 2- Apply the following concepts.

Alternate angles are equal to each other.

The addition of all angles of a triangle is 180.

Rhombus’s diagonals are perpendicular.

Two angles of the rhombus bisected by each diagonal of a rhombus.

the parallelogram is a rhombus if two consecutive sides of a parallelogram are equal.

Step 3- Step description.

Consider the quadrilateral $P R O V$ .

Since the opposite sides of a quadrilateral are parallel then it is a parallelogram.

Here consider that $\bar{P R} | | \bar{V O}; \bar{R O} | | \bar{P V}$ thus $P R O V$ is a parallelogram.

Also $\bar{P R} ≅ \bar{R O}$ thus the parallelogram $P R O V$ is a rhombus.

Step 4- Step description

From the figure $\bar{R V} ⊥ \bar{P O}$ , so that $∠ R E O = 90^{\circ}$ .

Consider the triangle $∠ R E O$ .

Since the sum of all the angles in a triangle is 180 degrees thus

$\begin{array}{l} m ∠ R E O + m ∠ E O R + m ∠ O R E = 180^{\circ} \\ 90 + m ∠ E O R + m ∠ 2 = 180 \\ m ∠ E O R + m ∠ 2 = 180 - 90 \\ m ∠ E O R + m ∠ 2 = 90 (1) \end{array}$

Also, from the figure $\bar{P R} | | \bar{V O}; \bar{R O} | | \bar{P V}$ and $\bar{R V}$ is a transversal therefore

$\begin{array}{l} m ∠ P R V ≅ m ∠ R V O \\ m ∠ P R V = m ∠ 1 (2) \end{array}$

The diagonal $\bar{R V}$ bisects the angle $∠ P R O$ .

That is $m ∠ P R V = m ∠ V R O (3)$

Use (1), (2) and (3) as follows:

$m ∠ 1 + m ∠ 2 = 90$

Therefore, the angles are complementary.

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For the following figure, given that $\bar{P R} | | \bar{V O}; \bar{R O} | | \bar{P V}; \bar{P R} ≅ \bar{R O}$ . Prove that $∠ 1$ and $∠ 2$ are complementary.

Short Answer

Step by step solution

Step 1- Check the figure.

Step 2- Apply the following concepts.

Step 3- Step description.

Step 4- Step description

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For the following figure, given thatPR¯||VO¯;RO¯||PV¯;PR¯≅RO¯ . Prove that∠1and∠2are complementary.

Short Answer

Step by step solution

Step 1- Check the figure.

Step 2- Apply the following concepts.

Step 3- Step description.

Step 4- Step description

One App. One Place for Learning.

Most popular questions from this chapter

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For the following figure, given that $\bar{P R} | | \bar{V O}; \bar{R O} | | \bar{P V}; \bar{P R} ≅ \bar{R O}$ . Prove that $∠ 1$ and $∠ 2$ are complementary.