1. Bisectors in Triangles
Lesson 5-2
Lesson 5-1 Quiz – Midsegments of Triangles
In GHI, R, S, and T are midpoints.
1. Name all the pairs of parallel segments.
2. If GH = 20 and HI = 18, find RT.
3. If RH = 7 and RS = 5, find ST.
4. If m G = 60 and m I = 70, find m GTR.
5. If m H = 50 and m I = 66, find m ITS.
6. If m G = m H = m I and RT = 15, find the perimeter of GHI.
RT || HI, RS || GI, ST || HG 9 7 70 64 90
5-1

2. When a point is the same distance from two or more
Bisectors in Triangles
Lesson 5-2
When a point is the same distance from two or more
objects, the point is said to be equidistant from
the objects.
Triangle congruence theorems can be
used to prove theorems about equidistant points.
5-2

3. Bisectors in Triangles
Lesson 5-2
5-2

4. Bisectors in Triangles
Lesson 5-2
5-2

5. Bisectors in Triangles
Lesson 5-2
The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line.
5-2

6. Bisectors in Triangles
Lesson 5-2
5-2

7. Bisectors in Triangles
Lesson 5-2
Additional Examples
Real-World Connection
Use the map of Washington, D.C. Describe the set of points that are equidistant from the Lincoln Memorial and the Capitol.
The Converse of the Perpendicular Bisector Theorem states If a point
is equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of the segment.
Quick Check
5-2

8. Bisectors in Triangles
Lesson 5-2
Additional Examples
(continued)
A point that is equidistant from the Lincoln Memorial and the Capitol
must be on the perpendicular bisector of the segment whose
endpoints are the Lincoln Memorial and the Capitol.
Therefore, all points on the perpendicular bisector of the segment
whose endpoints are the Lincoln Memorial and the Capitol are
equidistant from the Lincoln Memorial and the Capitol.
Quick Check
5-2

9. Bisectors in Triangles
Lesson 5-2
Additional Examples
Using the Angle Bisector Theorem
Find x, FB, and FD in the diagram above.
FD = FB Angle Bisector Theorem
7x – 37 = 2x Substitute.
7x = 2x Add 37 to each side.
5x = Subtract 2x from each side.
x = Divide each side by 5.
FB = 2(8.4) + 5 = Substitute.
FD = 7(8.4) – 37 = Substitute.
Quick Check
5-2

10. Bisectors in Triangles
Lesson 5-2
Lesson Quiz
Use this figure for Exercises 1–3.
1. Find BD.
2. Complete the statement:
C is equidistant from ? .
3. Can you conclude that
CN = DN? Explain.
Use this figure for Exercises 4–6.
4. Find the value of x.
5. Find CG.
6. Find the perimeter of
quadrilateral ABCG. 16 6 8
points A and B
No; if CN = DN, CNB DNB by SAS and CB = DB by CPCTC, which
is false. 48
5-2

11. Bisectors in Triangles
Lesson 5-2
Check Skills You’ll Need
1. Draw a triangle, XYZ. Construct STV so that STV XYZ.
2. Draw acute  P. Construct  Q so that
3. Draw AB. Construct a line CD so that CD AB and CD bisects AB.
4. Draw acute angle  E. Construct the bisector of  E.
TM bisects  STU so that m STM = 5x + 4 and m MTU = 6x – 2.
5. Find the value of x.
6. Find m STU.
(For help, go to Lesson 1-7.)
Use a compass and a straightedge for the following.
 Q  P.
Check Skills You’ll Need
5-2

12. Bisectors in Triangles
Lesson 5-2
Check Skills You’ll Need
Solutions
1-4. Answers may vary. Samples given:
5. Since TM bisects  STU, m STM = m MTU. So, 5x + 4 = 6x – 2.
Subtract 5x from both sides: 4 = x – 2; add 2 to both sides: x = 6.
6. From Exercise 5, x = 6. m STU = m STM + m MTU = 5x x – 2
= 11x + 2 = 11(6) + 2 = 68.
5-2