4.8 Use Isosceles and Equilateral Triangles
• You will use theorems about isosceles and equilateral triangles.
• Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles?
You will learn how to answer this question by learning the Base Angles Theorem and its converse.
Warm-Up ExercisesEXAMPLE 1 Apply the Base Angles Theorem
SOLUTION
In DEF, DE DF . Name two congruent angles.
DE DF , so by the Base Angles Theorem, E F.
Warm-Up ExercisesGUIDED PRACTICE for Example 1
SOLUTION
Copy and complete the statement.
1. If HG HK , then ? ? .
HGK HKG
Warm-Up ExercisesGUIDED PRACTICE for Example 1
Copy and complete the statement.
If KHJ KJH, then ? ? .If KHJ KJH, then ? ? .2. 2.
SOLUTION
If KHJ KJH, then , KH KJ
Warm-Up ExercisesEXAMPLE 2 Find measures in a triangle
Find the measures of P, Q, and R.
The diagram shows that PQR is equilateral. Therefore, by the Corollary to the Base Angles Theorem, PQR is equiangular. So, m P = m Q = m R.
3(m P) = 180o
Triangle Sum Theorem
m P = 60o
Divide each side by 3.
The measures of P, Q, and R are all 60° .
ANSWER Can an equilateral triangle have anangle of 61?
Warm-Up ExercisesGUIDED PRACTICE for Example 2
3. Find ST in the triangle at the right.
SOLUTION
STU is equilateral, then its is equiangular
Thus ST = 5 ( Base angle theorem )
ANSWER
Warm-Up ExercisesGUIDED PRACTICE for Example 2
4. Is it possible for an equilateral triangle to have an angle measure other than 60°? Explain.
SOLUTION
No; it is not possible for an equilateral triangle to have angle measure other then 60°. Because the triangle sum theorem and the fact that the triangle is equilateral guarantees the angle measure 60° because all pairs of angles could be considered base of an isosceles triangle
Warm-Up ExercisesEXAMPLE 3 Use isosceles and equilateral triangles
ALGEBRA
Find the values of x and y in the diagram.
SOLUTION
STEP 2 Find the value of x. Because LNM LMN, LN LM and LMN is isosceles. You also know that LN = 4 because KLN is equilateral.
STEP 1 Find the value of y. Because KLN is equiangular, it is also equilateral and KN KL . Therefore, y = 4.
Explain how you could findm ∠ M.
Warm-Up ExercisesEXAMPLE 3 Use isosceles and equilateral triangles
LN = LM Definition of congruent segments
4 = x + 1 Substitute 4 for LN and x + 1 for LM.
3 = x Subtract 1 from each side.
Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem
Lifeguard Tower
In the lifeguard tower, PS QR and QPS PQR.
QPS PQR?
a. What congruence postulate can you use to prove that
b. Explain why PQT is isosceles.
c. Show that PTS QTR.
Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem
SOLUTION
Draw and label QPS and PQR so that they do not overlap. You can see that PQ QP , PS QR , and QPS PQR. So, by the SAS Congruence Postulate,
a.
QPS PQR.
b. From part (a), you know that 1 2 because corresp. parts of are . By the Converse of the Base Angles Theorem, PT QT , and
PQT is isosceles.
Warm-Up ExercisesEXAMPLE 4 Solve a multi-step problem
c. You know that PS QR , and 3 4 because corresp. parts of are . Also, PTS QTR by the Vertical Angles Congruence Theorem. So, PTS QTR by the AAS Congruence Theorem.
Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4
5. Find the values of x and y in the diagram.
SOLUTION
y° = 120°
x° = 60°
Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4
SOLUTION
QPS PQR. Can be shown by segment addition postulate i.e
a. QT + TS = QS and PT + TR = PR
6. Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that PTS QTR
Warm-Up ExercisesGUIDED PRACTICE for Examples 3 and 4
Since PT QT from part (b) and
TS TR from part (c) then,
QS PR
PQ PQ Reflexive Property and
PS QR Given
Therefore QPS PQR . By SSS Congruence Postulate
ANSWER
Warm-Up ExercisesDaily Homework Quiz
Find the value of x.
1.
ANSWER 8
Warm-Up ExercisesDaily Homework Quiz
Find the value of x.
2.
ANSWER 3
Warm-Up ExercisesDaily Homework Quiz
If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles?
3.
ANSWER 34°, 34°
Warm-Up ExercisesDaily Homework Quiz
Find the perimeter of triangle.4.
ANSWER 66 cm
• You will use theorems about isosceles and equilateral triangles.
• Essential Question: How are the sides and angles of a triangle related if there are two or more congruent sides or angles?
• Angles opposite congruent sidesof a triangle are congruent andconversely.• If a triangle is equilateral, then it is equiangular and conversely.
If two sides of a triangle are congruent, then the angles opposite them are congruent. The converse is also true.