Download Mod 1 - Aim #22 - Manhasset Schools

Aim #22: How are base angle properties of an isosceles triangle CC Geometry H
used in proofs?
Do Now: State additional pieces of information needed to prove that each
pair of triangles are congruent by the SAS triangle congruence criteria.
1. Given: AB ≅ DC
Prove: ΔABC ≅ ΔDCB
2. Given: AB ≅ RS, AB || RS
Prove: ΔABC ≅ ΔRST
3a) If m≮EBC = 110 and m≮BCF = 110, find m≮ABE and m≮DCF.
F
E
b) What do you notice about your answers in (a)?
B
A
C
c) What can we conclude about supplements of equal angles?
Isosceles Triangle
Definition: If a triangle is isosceles, then it has at least 2
≅ sides.
I. Prove: "Base angles of an isosceles triangles are congruent", using SAS
Start with an isosceles Δ
with two ≅ sides.
Draw AD, the
bisector of ≮BAC
Since AD ≅ AD,
ΔABD ≅ ΔACD
because of SAS
D
II. Prove: "Base angles of an isosceles triangle are congruent", using transformations
Given: Isosceles ΔABC, with AB = AC
Prove: m B = m C
1. Draw the angle bisector AD which intersects BC at D.
We will demonstrate that rigid motions map point B to point C, and point C to
point B, by showing (1) AB maps to AC
and (2) AB = AC.
2. Since A is on the line of reflection (AD), rAD(A) = A.
Reflections preserve angle measure, so m≮BAD (after reflection) = m≮CAD.
Therefore, rAD(AB) = AC.
3. Reflections preserve lengths of segments and AB = AC, so rAD(AB) = AC
Therefore, rAD(B) = C. In a similar way, we can show rAD(C) = B.
4. Reflections map rays to rays, so r(BA) = CA and r(CA) = BA.
Since reflections preserve angle measures, the measure ofADr(≮ABC) = m≮ACB.
5. We conclude that m≮B = m≮C.
In proofs, we can state that
“Base angles of an isosceles triangle are equal in measure.”
or
“Base angles of an isosceles triangle are congruent.”
or
"If two sides of a triangle are equal, the angles opposite are equal."
Based on the diagram below, for each of the following, if the given congruence exists,
name the isosceles triangle and name the pair of congruent angles for the triangle.
Triangle
Congruent Angles
A
G
N
a) LE ≅ LG
b) EN ≅ NG
E
L
Exercises: Complete the following proofs.
J
1. Given: JK ≅ JL; JR bisects KL.
Prove: JR
KL
K
Statements
R
Reasons
A
2. Given: AB ≅ AC; XB ≅XC.
Prove: AX bisects ≮BAC
X
B
Statements
C
Reasons
L
3. Given: JX≅ JY; KX ≅ LY
Prove: ΔJKL is isosceles
Statements
Reasons
4. Given: ΔABC, with XY the angle bisector of≮BYA, and BC || XY
Prove: YB ≅ YC
Statements
Reasons
CC Geometry H
HW #22
Name __________________
Date _________________
1.
A
G
N
E
L
Based on the diagram above, for each of the following, if the given congruence exists,
name the isosceles triangle and name the pair of congruent angles for the triangle.
Congruent Angles
Triangle
a) AE ≅ EL
b) AN ≅ LN
c) NG ≅ GL
d) AE ≅ EN
Exercises: Complete the following proofs using the SAS criteria for triangle congruence.
2. Given: AB ≅ BC; AD ≅ DC
B
Prove: Δ ADB and Δ CDB are right triangles.
Statements
Reasons
A
D
C
OVER
A
3. Given: AC ≅ AE; BF || CE
Prove: AB ≅ AF
B
C
Statements
F
D
E
Reasons
Review
4. Complete the table based on the series of rigid motions performed on ΔABC.
m
A
A'
B
B'
C'
A"
C"
B"
C'''
B'''
l
Sequence of rigid
motions (3)
Composition in
function notation
Sequence of
corresponding
sides
Sequence of
corresponding
angles
A'''
Triangle
congruence
statement