Download geometry module 1 lesson 29 special lines in

GEOMETRY
MODULE 1 LESSON 29
SPECIAL LINES IN TRIANLGES
OPENING EXERCISE
A. Use your compass and straightedge to find the midpoints of ̅̅̅̅
𝐴𝐵 and ̅̅̅̅
𝐴𝐶 . Label the midpoint
as X and Y, respectively.
̅̅̅̅.
B. Draw the midsegment 𝑋𝑌
The midsegment is a line segment that joins the midpoints of two sides of a triangle or
trapezoid.

Compare ∠𝐴𝑋𝑌 and ∠𝐴𝐵𝐶. Compare ∠𝐴𝑌𝑋 and ∠𝐴𝐶𝐵. Without using a protractor, what
would you guess is the relationship between these two pairs of angles?
𝑚∠𝐴𝑋𝑌 = 𝑚∠𝐴𝐵𝐶
𝑚∠𝐴𝑌𝑋 = 𝑚∠𝐴𝐶𝐵

̅̅̅̅ ?
From your conclusion above, what can now be said about ̅̅̅̅
𝑋𝑌 and 𝐵𝐶
̅̅̅̅
̅̅̅̅ ∥ 𝐵𝐶
𝑋𝑌
HOMEWORK QUESTIONS
Problem Set Module 1 Lesson 28 is due Thursday.
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DISCUSSION
̅̅̅̅ and 𝐴𝐶
̅̅̅̅ , we could have chosen any
Even though we chose to determine the midsegment of 𝐴𝐵
two sides to work with. Let’s now focus on the properties associated with a midsegment.
The midsegment of a triangle is parallel to the third side of the triangle and half the length of the
third side of the triangle.
Given: ̅̅̅̅
𝑋𝑌 is a midsegment of ∆𝐴𝐵𝐶.
̅̅̅̅ and 𝑋𝑌 = 1 𝐵𝐶
̅̅̅̅ ∥ 𝐵𝐶
Prove: 𝑋𝑌
2
̅̅̅̅ to point G so that 𝑋𝑌 = 𝑌𝐺. Draw 𝐺𝐶.
Construction: Extend 𝑋𝑌
STEP
JUSTIFICATION
1
̅̅̅̅ is a midsegment of ∆𝐴𝐵𝐶
𝑋𝑌
Given
2
𝑋𝑌 = 𝑌𝐺
Construction
3
𝑚∠𝐴𝑌𝑋 = 𝑚∠𝐺𝑌𝐶
Vertical Angles
4
𝐴𝑌 = 𝑌𝐶
Property of Midpoint
5
∆𝐴𝑋𝑌 ≅ ∆𝐶𝐺𝑌
SAS
6
𝐺𝐶 = 𝐴𝑋
Corresponding sides of congruent triangles are equal in length.
7
𝐺𝐶 = 𝐵𝑋
Substitution (Line 1: 𝐴𝑋 = 𝐵𝑋)
8
𝑚∠𝐴𝑋𝑌 = 𝑚∠𝐶𝐺𝑌
Corresponding angles of congruent triangles are equal in measure.
9
̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐺𝐶
Alternate Interior Angles Converse
10
BXGC is a parallelogram
11
12
𝑋𝐺 = 𝐵𝐶
̅̅̅̅
̅̅̅̅
𝑋𝑌 ∥ 𝐵𝐶
𝑋𝐺 = 2𝑋𝑌
Substitution (Line #2 and 𝑋𝑌 + 𝑌𝐺 = 𝑋𝐺)
13
𝐵𝐶 = 2𝑋𝑌
Substitution (Line #11)
14
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𝑋𝑌 =
1
𝐵𝐶
2
One pair of opposite sides is equal and parallel. (Line 7 and 9)
Property of Parallelogram
Division
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PRACTICE
Apply what you know about the properties of midsegments to solve the following exercises.
1.
a. Find the length of x.
15
b. Find the perimeter of ∆𝐴𝐵𝐶.
68
2.
a. Find the angle measure of x.
50°
b. Find the angle measure of y.
70°
3. In ∆𝑅𝑆𝑇, the midpoints of each side have been marked by points X, Y, and Z.

Mark the halves of each side divided by the midpoint with a
congruency mark.

Draw midsegments ̅̅̅̅
𝑋𝑌, ̅̅̅̅
𝑌𝑍, and ̅̅̅̅
𝑋𝑍. March each
midsegment with the appropriate congruency mark for the
sides of the triangle.
a. What conclusion can you draw about the four triangles within ∆𝑅𝑆𝑇? Why?
All four triangles are congruent by SSS.
b. State the appropriate correspondences among the four triangles within ∆𝑅𝑆𝑇.
∆𝑅𝑋𝑌, ∆𝑌𝑍𝑇, ∆𝑋𝑆𝑍, ∆𝑍𝑌𝑋
c. State a correspondence between ∆𝑅𝑆𝑇 and nay one of the four small triangles.
∆𝑅𝑋𝑌, ∆𝑅𝑆𝑇
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ON YOUR OWN
1. Find x.
̅̅̅̅̅ and ̅̅̅̅̅
2. R and S are the midpoints of 𝑋𝑊
𝑊𝑌, respectively. Find the perimeter of ∆𝑊𝑋𝑌.
SUMMARY
The midsegment of a triangle is parallel to the third side of the triangle and half the length of the
third side of the triangle.
1. 79° 2. 82
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