Problem 2 Making Conclusions From a Diagram

What can you conclude from the information in the diagram?

• ∠ 1 ≅ ∠ 2 angle 1 approximately equal to angle 2  by the markings.
• ∠ 3 angle 3  and ∠ 5 angle 5  are vertical angles.
• ∠ 1 angle 1  and ∠ 2 , ∠ 2 angle 2 comma angle 2  and ∠ 3 , ∠ 3 angle 3 comma angle 3  and ∠ 4 , ∠ 4 angle 4 comma angle 4  and ∠ 5 , angle 5 comma  and ∠ 5 angle 5  and ∠ 1 angle 1  are adjacent angles.
• ∠ 3 angle 3  and ∠ 4 , angle 4 comma  and ∠ 4 angle 4  and ∠ 5 angle 5  are adjacent supplementary angles. So, m ∠ 3 + m ∠ 4 = 180 m angle 3 plus m angle 4 equals 180  and m ∠ 4 + m ∠ 5 = 180 m angle 4 plus m angle 5 equals 180  by the definition of supplementary angles.

✓ Got It?

2. Can you make each conclusion from the information in the diagram? Explain.

   

   1. T W ¯ ≅ W V ¯ t w bar , approximately equal to , w v bar
   2. P W ¯ ≅ W Q ¯ p w bar , approximately equal to , w q bar
   3. ∠ TWQ angle  is a right angle.
   4. T V ¯ t v bar  bisects P Q ¯ . p q bar , .

Take note Postulate 1-9: Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Problem 3 Finding Missing Angle Measures

Algebra ∠ KPL angle  and ∠ JPL angle  are a linear pair, m ∠ KPL = 2 x + 24 , bold italic m angle  and m ∠ JPL = 4 x + 36 . bold italic m angle  What are the measures of ∠ KPL angle  and ∠ JPL ? angle

• Step 1

  m ∠ KPL + m ∠ JPL = 180 Def. of supplementary angles ( 2 x + 24 ) + ( 4 x + 36 ) = 180 Substitute. 6 x + 60 = 180 Combine like terms. 6 x = 120 Subtract  60  from each side. x = 20 Divide each side by  6 . table with 5 rows and 3 columns , row1 column 1 , m angle

  

• Step 2 Evaluate the original expressions for x = 20 . x equals 20 .

  m ∠ KPL = 2 x + 24 = 2 · 20 + 24 = 40 + 24 = 64 Substitute  20  for  x . m ∠ JPL = 4 x + 36 = 4 · 20 + 36 = 80 + 36 = 116 table with 2 rows and 3 columns , row1 column 1 , m angle