You will use theorems about isosceles and equilateral triangles.
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SOLUTION
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If HG HK , then ? ? .
If KHJ KJH, then ? ? .
If KHJ KJH, then ? ? .
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Can an equilateral triangle have an
angle of 61?
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)
The measures of P, Q, and R are all 60° .
ANSWER
STU is equilateral, then its is equiangular
Thus ST = 5
( Base angle theorem )
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
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SOLUTION
m ∠ M.
Find the values of x and y in the diagram.
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
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LN = LM
3 = x
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QPS PQR?
Explain why PQT is isosceles.
Show that PTS QTR.
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,
QPS PQR.
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.
Solve a multi-step problem
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.
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SOLUTION
Find the values of x and y in the diagram.
y° = 120°
x° = 60°
SOLUTION
QPS PQR. Can be shown by segment addition postulate i.e
a.
QT + TS = QS and PT + TR = PR
Use parts (b) and (c) in Example 4 and the SSS Congruence Postulate to give a different proof that
PTS QTR
Since PT QT from part (b) and
from part (c) then,
ANSWER
1.
ANSWER
8
2.
ANSWER
3
Daily Homework Quiz
If the measure of vertex angle of an isosceles triangle is 112°, what are the measures of the base angles?
3.
ANSWER
4.
ANSWER
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 sides
conversely.
• If a triangle is equilateral, then it is equiangular and conversely.
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