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 (linear pair) 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:

Step1: Recall Definition of Congruence

The definition of congruent angles states that if two angles are congruent, their measures are equal. So if \( \angle D \cong \angle E \), then \( m\angle D = m\angle E \).

Step2: Recall Definition of Complementary Angles

Complementary angles are two angles whose measures add up to \( 90^\circ \). So if \( m\angle 1 + m\angle 2 = 90^\circ \), then \( \angle 1 \) and \( \angle 2 \) are complementary angles.

Step3: Recall Definition of Supplementary Angles

Supplementary angles are two angles whose measures add up to \( 180^\circ \). So if \( \angle P \) and \( \angle Q \) are supplementary angles, then \( m\angle P + m\angle Q = 180^\circ \).

Step4: Recall Definition of a Right Angle

A right angle is an angle whose measure is \( 90^\circ \). So if \( m\angle JKL = 90^\circ \), then \( \angle JKL \) is a right angle.

Step5: Recall Vertical Angles Theorem

The Vertical Angles Theorem states that vertical angles are congruent. So if \( \angle 3 \) and \( \angle 4 \) are vertical angles, then \( \angle 3 \cong \angle 4 \) (or \( m\angle 3 = m\angle 4 \)).

Step6: Recall Complement Theorem

The Complement Theorem states that if two angles form a right angle (i.e., their non - common sides form a right angle), then they are complementary. So if \( \angle S \) and \( \angle T \) form a right angle, then \( \angle S \) and \( \angle T \) are complementary (or \( m\angle S + m\angle T = 90^\circ \)).

Step7: Recall Supplement (Linear Pair) Theorem

A linear pair of angles is supplementary, meaning their measures add up to \( 180^\circ \). So if \( \angle X \) and \( \angle Y \) form a linear pair, then \( m\angle X + m\angle Y = 180^\circ \) (or \( \angle X \) and \( \angle Y \) are supplementary).

Step8: Recall Congruent Complements Theorem

The Congruent Complements Theorem states that if two angles are complementary to the same angle (or congruent angles), then they are congruent. So if \( \angle 1 \) is complementary to \( \angle 2 \) and \( \angle 2 \) is complementary to \( \angle 4 \), then \( \angle 1 \cong \angle 4 \) (or \( m\angle 1 = m\angle 4 \)).

Step9: Recall Congruent Supplements Theorem

The Congruent Supplements Theorem states that if two angles are supplementary to the same angle (or congruent angles), then they are congruent. So if \( \angle J \) is supplementary to \( \angle K \) and \( \angle J \) is supplementary to \( \angle L \), then \( \angle K \cong \angle L \) (or \( m\angle K = m\angle L \)).

Answer:

s:

1. \( m\angle D = m\angle E \)
2. \( \angle 1 \) and \( \angle 2 \) are complementary angles (or \( \angle 1 \) is complementary to \( \angle 2 \))
3. \( m\angle P + m\angle Q = 180^\circ \)
4. \( \angle JKL \) is a right angle
5. \( \angle 3 \cong \angle 4 \) (or \( m\angle 3 = m\angle 4 \))
6. \( \angle S \) and \( \angle T \) are complementary (or \( m\angle S + m\angle T = 90^\circ \))
7. \( m\angle X + m\angle Y = 180^\circ \) (or \( \angle X \) and \( \angle Y \) are supplementary)
8. \( \angle 1 \cong \angle 4 \) (or \( m\angle 1 = m\angle 4 \))
9. \( \angle K \cong \angle L \) (or \( m\angle K = m\angle L \))