Question

use the given information to prove that ( mangle 3 = mangle 4 ).
given: ( mangle ) partially obscured ( mangle 1 ) partially obscured
prove: ( mangle 3 ) partially obscured
step statement
1 ( mangle qps = mangle tpv )
2 ( mangle 1 = mangle 2 )
3 ( mangle 3 = mangle 4 )
select the rule

• given
• reflexive property
• transitive property
• addition and subtraction properties
• multiplication and division properties
• substitution property
• definition of congruent segments

Explanation:

Step1: Identify given equal angles

$m\angle QPS = m\angle TPV$; $m\angle 1 = m\angle 2$

Step2: Decompose the larger angles

$m\angle QPS = m\angle 1 + m\angle 3$; $m\angle TPV = m\angle 2 + m\angle 4$

Step3: Substitute equal angles

Substitute $m\angle QPS$ and $m\angle TPV$: $m\angle 1 + m\angle 3 = m\angle 2 + m\angle 4$

Step4: Subtract equal angles

Subtract $m\angle 1$ (=$m\angle 2$) from both sides, using Subtraction Property (a subset of Addition and Subtraction Properties):
$m\angle 1 + m\angle 3 - m\angle 1 = m\angle 2 + m\angle 4 - m\angle 2$
Simplify to get $m\angle 3 = m\angle 4$

Answer:

For Step 1 (Statement: $m\angle QPS = m\angle TPV$): Given
For Step 2 (Statement: $m\angle 1 = m\angle 2$): Given
For Step 3 (Statement: $m\angle 3 = m\angle 4$): Addition and Subtraction Properties