Question

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given each definition or theorem, complete each statement.

1. definition of congruence:

if $\angle d \cong \angle e$, then

1. definition of complementary angles:

if $m\angle 1 + m\angle 2 = 90^\circ$, then

1. definition of supplementary angles:

if $\angle p$ and $\angle q$ are supplementary angles, then

1. definition of a right angle:

if $m\angle jkl = 90^\circ$, then

1. vertical angles theorem:

if $\angle 3$ and $\angle 4$ are vertical angles, then

1. complement theorem:

if $\angle s$ and $\angle t$ form a right angle, then

1. supplement theorem:

if $\angle x$ and $\angle y$ form a linear pair, then

1. congruent complements theorem: if $\angle 1$ is complementary to $\angle 2$

and $\angle 2$ is complementary to $\angle 4$, then

1. congruent supplements theorem: if $\angle j$ is supplementary to $\angle k$

and $\angle j$ is supplementary to $\angle l$, then

Explanation:

1. Definition of Congruence

Step1: Recall Congruence Definition

Congruent angles have equal measures. So if \( \angle D \cong \angle E \), then their measures are equal.

Step2: Formulate the Statement

Using the definition, we write \( m\angle D = m\angle E \).

Step1: Recall Complementary Angles

Complementary angles are two angles whose sum of measures is \( 90^\circ \). So if \( m\angle 1 + m\angle 2 = 90^\circ \), they are complementary.

Step2: Formulate the Statement

We state that \( \angle 1 \) and \( \angle 2 \) are complementary angles.

Step1: Recall Supplementary Angles

Supplementary angles are two angles whose sum of measures is \( 180^\circ \). So if \( \angle P \) and \( \angle Q \) are supplementary, their measures add to \( 180^\circ \).

Step2: Formulate the Statement

We write \( m\angle P + m\angle Q = 180^\circ \).

Answer:

\( m\angle D = m\angle E \)

2. Definition of Complementary Angles