Question

think back on the properties and theorems you explored in this lesson
1 choose the correct property or theorem for each statement.
① $angle h cong angle k$
$angle k cong angle m$
therefore, $angle h cong angle m$
② $moverline{mn} = moverline{op}$
$moverline{mn} + moverline{rs} = moverline{op} + moverline{rs}$
③ $mangle t = 34^circ$
$mangle w = 34^circ$
therefore, $mangle t = mangle w$
④ $mangle a + mangle b = 180^circ$
$mangle a + mangle c = 180^circ$
therefore, $angle b cong angle c$
⑤ $mangle v = mangle v$
⑥ $angle 1 = angle 2$
property or theorem
addition property of equality
congruent supplement theorem
reflexive property
substitution property
transitive property
vertical angle theorem

Explanation:

Part (a)

Step 1: Identify the property

The Transitive Property of Congruence (or Equality for angles) states that if \( \angle H \cong \angle K \) and \( \angle K \cong \angle M \), then \( \angle H \cong \angle M \). So we match this to the Transitive Property.

Step 2: Select the property

From the list, the Transitive Property is the correct one for \( \angle H \cong \angle K \), \( \angle K \cong \angle M \), therefore \( \angle H \cong \angle M \).

Part (b)

Step 1: Identify the property

The Addition Property of Equality states that if \( a = b \), then \( a + c = b + c \). Here \( m\overline{MN}=m\overline{OP} \), so \( m\overline{MN}+m\overline{RS}=m\overline{OP}+m\overline{RS} \) follows the Addition Property of Equality.

Step 2: Select the property

So we choose the Addition Property of Equality.

Part (c)

Step 1: Identify the property

The Substitution Property allows us to substitute equal values. Since \( m\angle T = 34^\circ \) and \( m\angle W = 34^\circ \), we can say \( m\angle T=m\angle W \) by substituting the equal measures.

Step 2: Select the property

So the Substitution Property is correct.

Part (d)

Step 1: Identify the theorem

The Congruent Supplement Theorem states that if two angles are supplements of the same angle (or congruent angles), then they are congruent. Here \( m\angle A + m\angle B = 180^\circ \) and \( m\angle A + m\angle C = 180^\circ \), so \( \angle B\cong\angle C \) by the Congruent Supplement Theorem.

Step 2: Select the theorem

So we choose the Congruent Supplement Theorem.

Part (e)

Step 1: Identify the property

The Reflexive Property states that any quantity is equal (or congruent) to itself. So \( m\angle V = m\angle V \) is an example of the Reflexive Property.

Step 2: Select the property

So we choose the Reflexive Property.

Part (f)

Step 1: Identify the theorem

The Vertical Angle Theorem states that vertical angles are congruent. In the diagram, \( \angle 1 \) and \( \angle 2 \) are vertical angles, so \( \angle 1=\angle 2 \) by the Vertical Angle Theorem.

Step 2: Select the theorem

So we choose the Vertical Angle Theorem.

Answer:

(a) Transitive Property
(b) Addition Property of Equality
(c) Substitution Property
(d) Congruent Supplement Theorem
(e) Reflexive Property
(f) Vertical Angle Theorem