Question

1. given: ∠p and ∠r are supplementary and ∠q and ∠p are supplementary. prove: ∠q ≅ ∠r.

statement | reason
1 | ∠p and ∠r are supplementary
2 | given
3 | m∠p + m∠r = 180°
4 | definition of supplementary angles
5 | m∠p + m∠r = m∠q + m∠p
6 | transitive property of equality
7 | m∠q = m∠r
8 | ∠q ≅ ∠r

definition of supplementary angles definition of congruent angles ∠q and ∠p are supplementary given m∠q + m∠p = 180° subtraction property of equality

1. complete the following two-column proof.

given $overline{rs} \cong \overline{ps}$ and $overline{st} \cong \overline{sq}$

prove $overline{rt} \cong \overline{pq}$

diagram: r---s---t and q---s---p, with s as intersection, segments rs, ps, st, sq marked congruent

statements | reasons

1. $overline{rs} \cong \overline{ps}$; $overline{st} \cong \overline{sq}$ | 1. given
2. $rs = ps$; $st = sq$ | 2. _______________
3. $rs + st = rt$; $ps + sq = pq$ | 3. _______________
4. _______________ | 4. substitution property of equality
5. $overline{rt} \cong \overline{pq}$ | 5. _______________

box with options: rt = ps; rt = pq; segment addition postulate; symmetric property of equality; substitution property of equality; definition of congruent segments

Explanation:

Problem 1

Step1: Fill Statement 2

Statement 2 should be the other given, so $\angle Q$ and $\angle P$ are supplementary.

Step2: Fill Reason 6

From $m\angle P + m\angle R = m\angle Q + m\angle P$, subtract $m\angle P$ from both sides (Subtraction Property of Equality) to get $m\angle Q = m\angle R$.

Step3: Fill Reason 7

If $m\angle Q = m\angle R$, by Definition of congruent angles, $\angle Q \cong \angle R$.

Step2: Reason for $RS = PS; ST = SQ$

By Definition of congruent segments (if $\overline{RS} \cong \overline{PS}$, then $RS = PS$ and same for $\overline{ST} \cong \overline{SQ}$).

Step3: Reason for $RS + ST = RT$ and $PS + SQ = PQ$

This is the Segment Addition Postulate (the sum of two adjacent segments equals the whole segment).

Step4: Statement for Substitution

Substitute $RS = PS$ and $ST = SQ$ into $RS + ST = RT$ and $PS + SQ = PQ$ to get $RT = PQ$.

Step5: Reason for $\overline{RT} \cong \overline{PQ}$

By Definition of congruent segments (if $RT = PQ$, then $\overline{RT} \cong \overline{PQ}$).

Answer:

• Statement 2: $\boldsymbol{\angle Q}$ and $\boldsymbol{\angle P}$ are supplementary
• Reason 6: Subtraction Property of Equality
• Reason 7: Definition of congruent angles

Problem 2