Exercise 19 Page 578

We are given a diagram showing △KLM and its midsegments JH, JP, and PH. Let's find the value of x.

Since ∠ LHM is a straight angle, and we already know the measures of ∠ MHP and ∠ JHL. We can calculate the measure of ∠ PHJ by using the Angle Addition Postulate.

m∠ MHP + m∠ PHJ + m∠ JHL = 180 °

SubstituteII

m∠ MHP= 44 °, m∠ JHL= 76 °

44 ° + m∠ PHJ + 76 ° = 180 °

▼

Solve for m∠ PHJ

AddTerms

Add terms

m∠ PHJ + 120 ° = 180 °

SubEqn

LHS-120 °=RHS-120 °

m∠ PHJ = 60 °

The Triangle Midsegment Theorem tells us that if a segment joins the midpoints of two sides of a triangle, then not only is it parallel to the third side of the triangle, but also its length is half the length of that side. Since P, J, and H are the midpoints of the sides of the triangle, we can note some parallel segments. JP ∥ LM JH ∥ KM PH ∥ KL According to the Alternate Interior Angles Theorem, since JH cuts the parallel segments PH and KL, then ∠ PHJ and ∠ LJH are congruent angles. Let's use this information to find x. m∠ PHJ = m∠ LJH ⇕ 60 ° = x ° Therefore, we found that x=60.