Question

The diagram,

Given:$\angle 5$is supplementary to $\angle 4$.

a. What can you conclude about$\angle 5$and $\angle 3$.

b. State the theorem that justifies the conclusion.

Short answer

a. $\angle 5$and $\angle 3$are congruent.

b. The conclusion is justified by the concept of supplementary angles.

Step-by-step solution

Part a. Step 1. Observe the diagram.

Here, $\angle 5$is supplementary to$\angle 4$.

Part a. Step 2. State the concept used.

If angles formed by two intersecting lines are adjacent, the two angle are said to be linear.

From the figure, $\angle 3$and $\angle 4$form a linear pair, so their sum is $180°$.

Part a. Step 3. Show the prove.

Here, $\angle 5$is supplementary to $\angle 4$it is given.

Using the definition of supplementary angles that is if two angles are supplementary then their sum is $180°$.

So, $\angle 5+\angle 4=180°$

Again $\angle 4+\angle 3=180°$

Now by substitution property,

$\angle 5+\angle 4=\angle 4+\angle 3$

Therefore, $\angle 5=\angle 3$( by subtraction property).

Part a. Step 4. State the conclusion.

Therefore, $\angle 5$ and $\angle 3$ are congruent.

Part b. Step 1. Observe the diagram.

Here, $\angle 5$ is supplementary to $\angle 4$.

Part b. Step 2. Show the two-column proof.

If the sum of two angles is $180°$, then they are supplementary angles.

And if angles formed by two intersecting lines are adjacent, the two angle are said to be linear.

From the figure, $\angle 3$and $\angle 4$form a linear pair, so their sum is $180°$.

The two-column proof is :

Statement	Reasons
$\angle 5$is supplementary to$\angle 4$	Given.
$\angle 5+\angle 4=180°$	Definition of supplementary angles.
$\angle 4+\angle 3=180°$	Given.
$\angle 5+\angle 4=\angle 4+\angle 3$	Substitution Property.
$\angle 5=\angle 3$	Subtraction Property.

Part b. Step 3. State the conclusion.

Therefore, the concept of supplementary angles and linear pairs and using substitution property, $\angle 5$and$\angle 3$ are congruent