In the exercise we are given the following figure, including angles $1$ and $2 .$

We are asked to complete the following paragraph proof.

Given	$m ∠ 1 = m ∠ 2,$ $∠ 1$ and $∠ 2$ are supplementary.
Prove	$∠ 1$ and $∠ 2$ are right angles.
Proof	$m ∠ 1 + m ∠ 2 = \underline{w o r d}$ since they are supplementary angles. Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 1 + m ∠ 1 = 18 0^{\circ}$ by $\underline{w o r d w o r d} .$ Solving the equation gives $m ∠ 1 = \underline{w o r d} .$ Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 2$ is also $\underline{w o r d} .$ Therefore, $∠ 1$ and $∠ 2$ are right angles.

Let's do it! First, if two angles are supplementary, then by definition their angle measures add up to $18 0^{\circ} .$ We can fill in the first gap.

Given	$m ∠ 1 = m ∠ 2,$ $∠ 1$ and $∠ 2$ are supplementary.
Prove	$∠ 1$ and $∠ 2$ are right angles.
Proof	$m ∠ 1 + m ∠ 2 = \underline{18 0^{\circ}}$ since they are supplementary angles. Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 1 + m ∠ 1 = 18 0^{\circ}$ by $\underline{w o r d w o r d} .$ Solving the equation gives $m ∠ 1 = \underline{w o r d} .$ Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 2$ is also $\underline{w o r d} .$ Therefore, $∠ 1$ and $∠ 2$ are right angles.

Now, we can rewrite the second sentence of the proof using mathematical symbols.

m ∠ 1 = m ∠ 2 \Rightarrow m ∠ 1 + m ∠ 1 = 18 0^{\circ}

We see that

m ∠ 1

was substituted for

m ∠ 2

in the expression. The resulting statement is still true because the angle measures are equal.

Given	$m ∠ 1 = m ∠ 2,$ $∠ 1$ and $∠ 2$ are supplementary.
Prove	$∠ 1$ and $∠ 2$ are right angles.
Proof	$m ∠ 1 + m ∠ 2 = \underline{18 0^{\circ}}$ since they are supplementary angles. Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 1 + m ∠ 1 = 18 0^{\circ}$ by $\underline{substitution} .$ Solving the equation gives $m ∠ 1 = \underline{w o r d} .$ Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 2$ is also $\underline{w o r d} .$ Therefore, $∠ 1$ and $∠ 2$ are right angles.

To solve the resulting equation, we should isolate

m ∠ 1

on one side of the equation. Let's do it!

m ∠ 1 + m ∠ 1 = 18 0^{\circ}

Add terms

2 m ∠ 1 = 18 0^{\circ}

$LHS / 2 = RHS / 2$

m ∠ 1 = 9 0^{\circ}

We found that

m ∠ 1 = 9 0^{\circ} .

Let's write that down.

Given	$m ∠ 1 = m ∠ 2,$ $∠ 1$ and $∠ 2$ are supplementary.
Prove	$∠ 1$ and $∠ 2$ are right angles.
Proof	$m ∠ 1 + m ∠ 2 = \underline{18 0^{\circ}}$ since they are supplementary angles. Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 1 + m ∠ 1 = 18 0^{\circ}$ by $\underline{substitution} .$ Solving the equation gives $m ∠ 1 = \underline{9 0^{\circ}} .$ Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 2$ is also $\underline{w o r d} .$ Therefore, $∠ 1$ and $∠ 2$ are right angles.

Because angles $1$ and $2$ have the same measure, we know that $m ∠ 2$ is also $9 0^{\circ} .$

Given	$m ∠ 1 = m ∠ 2,$ $∠ 1$ and $∠ 2$ are supplementary.
Prove	$∠ 1$ and $∠ 2$ are right angles.
Proof	$m ∠ 1 + m ∠ 2 = \underline{18 0^{\circ}}$ since they are supplementary angles. Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 1 + m ∠ 1 = 18 0^{\circ}$ by $\underline{substitution} .$ Solving the equation gives $m ∠ 1 = \underline{9 0^{\circ}} .$ Since $m ∠ 1 = m ∠ 2,$ then $m ∠ 2$ is also $\underline{9 0^{\circ}} .$ Therefore, $∠ 1$ and $∠ 2$ are right angles.

Our paragraph proof is now complete.

Exercise