Question

Use the condition: Two angles are congruent if they are vertical angles.

a. Write the hypothesis.

b. Write the converse.

Short answer

a. Two angles are vertical angles, then they are congruent.

b. The converse statement “If two angles are congruent, they are vertical angles” is false.

Step-by-step solution

Part a. Step 1. State the definition of vertical angle.

When two lines intersect to make an X, angles on opposite sides of the X are called vertical angles. These angles are equal and here is the official theorem that tells so.

Part a. Step 2. Sketch the diagram.

Part a. Step 3. State the conclusion.

Vertical angles are congruent:

If two angles are vertical angles, then they are congruent (from the figure above).

Vertical angles are one of the most frequently used things in proofs and other types of geometry problems, and they are one of the easiest things to spot in a diagram.

Part b. Step 1. State the converse of the statement.

If two angles are congruent, they are vertical angles.

Part b. Step 2. State the explanation.

Two angles of an isosceles triangle have the same measure (are congruent) but they are not vertical angles. So, if two angles are congruent that does not mean that they are vertical angles.

Part b. Step 3. State the conclusion.

Therefore, the converse is false.