Question

∠x and ∠y form a linear pair
∠x ≅ ∠y
∠z ≅ ∠x
prove
∠z is a right angle
statement

1. ∠x and ∠y form a linear pair
2. ∠x and ∠y are supplementary
3. ∠x ≅ ∠y
4. ∠x is a right angle
5. ∠z ≅ ∠x
6. ∠z is a right angle

reason

1. given
2. definition of a linear pair
3. given
4. ?
5. given
6. congruent right angles theorem

what geometric reason justifies the transition from statement 2 and 3 to statement 4?
supplemental right angles theorem
congruent right angles theorem
substitution
vertical angles theorem

Explanation:

Step1: Recall the properties of supplementary angles

If two angles are supplementary, their sum is \(180^{\circ}\). So, \(\angle X+\angle Y = 180^{\circ}\) (from statement 2).

Step2: Use the congruent relationship

Since \(\angle X\cong\angle Y\) (statement 3), we can substitute \(\angle Y\) with \(\angle X\). Then \(\angle X+\angle X=180^{\circ}\), which simplifies to \(2\angle X = 180^{\circ}\).

Step3: Solve for \(\angle X\)

Dividing both sides of \(2\angle X = 180^{\circ}\) by 2 gives \(\angle X = 90^{\circ}\). This substitution (replacing \(\angle Y\) with \(\angle X\) because they are congruent) is what allows us to conclude that \(\angle X\) is a right - angle (statement 4).

Answer:

C. Substitution