Download congruent triangles

Chapters 4 and 5
Triangles
Which type of triangle is this?
Scalene Triangle
None of the sides and none
of the angles are the same.
Which type of triangle is this?
Isosceles Triangle
Two sides and two
angles are the same.
Which type of triangle is this?
Equilateral Triangle
All sides and all
angles are the same.
Opposite
A
Which side is opposite ∠𝑨?
𝑩𝑪
Which side is opposite ∠𝑩?
𝑨𝑪
Which side is opposite ∠𝑪? 𝑨𝑩
C
B
Congruent Triangles 4.1
Theorem 4.1: Triangle Sum Theorem- The sum of
all the interior angles of every triangle is 𝟏𝟖𝟎°.
60°
15°
100°
60°
60°
65°
Practice
Solve for x and y.
𝒚
𝟗𝒙 − 𝟏𝟐 + 𝟔𝒙 − 𝟑 + 𝟗𝟎 = 𝟏𝟖𝟎
−𝟗𝟎 −𝟗𝟎
𝟗𝒙 − 𝟏𝟐 + 𝟔𝒙 − 𝟑 = 𝟗𝟎
𝟏𝟓𝒙 − 𝟏𝟓 = 𝟗𝟎
+𝟏𝟓 +𝟏𝟓
𝟏𝟓𝒙 = 𝟏𝟎𝟓
𝟏𝟓
𝟏𝟓
𝟗𝒙 − 𝟏𝟐
𝒙=𝟕
𝟔𝒙 − 𝟑
𝟗𝒙 − 𝟏𝟐 + 𝒚 = 𝟏𝟖𝟎
𝟗 𝟕 − 𝟏𝟐 + 𝒚 = 𝟏𝟖𝟎
𝟓𝟏 + 𝒚 = 𝟏𝟖𝟎
−𝟓𝟏
−𝟓𝟏
𝒚 = 𝟏𝟐𝟗
Theorem 4.2: Triangle Exterior Angle Conjecture- The
measure of the exterior angle of a triangle is…
equal to the sum of the measures of the two
nonadjacent interior angles.
𝟖𝟖°
𝒙
The angles circled in
red are nonadjacent
interior angles for x.
𝟔𝟎°
𝟑𝟐°
𝒙 = 𝟏𝟒𝟖°
Which side is
opposite ∠𝑨𝑩𝑫 ?
𝑨𝑫
A
D
Which side is
opposite ∠𝑩𝑫𝑪 ?
𝑩𝑪
Which angle is
opposite 𝑨𝑩 ?
∠𝑨𝑫𝑩
B
C
Practice
Find the missing angle measures.
𝟏𝟖𝟎 − 𝟓𝟏 = 𝟐𝒙
𝟏𝟐𝟗 = 𝟐𝒙
𝟐
𝟐
𝟔𝟒. 𝟓 = 𝒙
𝟓𝟏°
Practice
Find the missing angle measure.
x
𝒙 = 𝟒𝟓
Which type of triangle is this?
𝟑𝟎°
Acute Isosceles
Which type of triangle is this?
9 CM
11 CM
7 CM
Right Scalene
Practice
Find the degree measure of the interior angles
of ∆𝑾𝒀𝒁.
∠𝑾 = (𝒙 + 𝟐𝟓) ∠𝑾 = (𝟏𝟕 + 𝟐𝟓) ∠𝑾 = 𝟒𝟐°
∠𝒀 = (𝟒𝒙 + 𝟏𝟏) ∠𝒀 = 𝟒(𝟏𝟕) + 𝟏𝟏 ∠𝒀 = 𝟕𝟗°
∠𝒁 = (𝟑𝒙 + 𝟖) ∠𝒁 = 𝟑 𝟏𝟕 + 𝟖
∠𝒁 = 𝟓𝟗°
𝒙 + 𝟐𝟓 + 𝟒𝒙 + 𝟏𝟏 + 𝟑𝒙 + 𝟖 = 𝟏𝟖𝟎
𝟖𝒙 + 𝟒𝟒 = 𝟏𝟖𝟎
−𝟒𝟒 −𝟒𝟒
𝟖𝒙 = 𝟏𝟑𝟔
𝟖
𝟖
𝒙 = 𝟏𝟕
How do we know our answer is correct?
Three angles in any triangle should add up to 𝟏𝟖𝟎°
∠𝑾 = 𝟒𝟐°
+
∠𝒀 = 𝟕𝟗°
+
∠𝒁 = 𝟓𝟗°
𝟏𝟖𝟎°
Practice
Find the degree measure of the interior angles
of ∆𝑨𝑩𝑪.
∠𝑨 = (𝟖𝒙 + 𝟕)
∠𝑨 = 𝟖(𝟓) + 𝟕)
∠𝑨 = 𝟒𝟕°
∠𝑩 = (𝟏𝟐𝒙 − 𝟏𝟕)
∠𝑪 = (𝟐𝟎𝒙 − 𝟏𝟎)
∠𝑩 = 𝟏𝟐 𝟓 − 𝟏𝟕
∠𝑪 = 𝟐𝟎 𝟓 − 𝟏𝟎
∠𝑩 = 𝟒𝟑°
∠𝑪 = 𝟗𝟎°
𝟖𝒙 + 𝟕 + 𝟏𝟐𝒙 − 𝟏𝟕 + 𝟐𝟎𝒙 − 𝟏𝟎 = 𝟏𝟖𝟎
𝟒𝟎𝒙 − 𝟐𝟎 = 𝟏𝟖𝟎
+𝟐𝟎 +𝟐𝟎
𝟒𝟎𝒙 = 𝟐𝟎𝟎
𝟒𝟎
𝟒𝟎
𝒙=𝟓
How do we know our answer is correct?
Three angles in any triangle should add up to 𝟏𝟖𝟎°
∠𝑨 = 𝟒𝟕°
+
∠𝑩 = 𝟒𝟑°
+
∠𝑪 = 𝟗𝟎°
𝟏𝟖𝟎°
Practice
Find the degree measure of the interior
angles of triangle ABC.
∠𝑨 = (𝒙 + 𝟖)
∠𝑩 = (𝟑𝒙 + 𝟕)
∠𝑪 = (𝟒𝒙 + 𝟓)
Review
Are the lines given by the following
equations perpendicular, parallel, or
neither?
𝒚 = 𝟕𝒙 − 𝟏𝟐
𝟏
𝒚=− 𝒙+𝟓
𝟕
Perpendicular
Review
Are the lines given by the following
equations perpendicular, parallel, or
neither?
𝒚 = 𝟐𝒙 − 𝟏𝟏
𝒚 = 𝟐𝒙 + 𝟗
Parallel
Review
Are the lines given by the following
equations perpendicular, parallel, or
neither?
𝟒
𝒚= 𝒙−𝟑
𝟓
Neither
𝟓
𝒚= 𝒙+𝟗
𝟒
Review
Are the lines given by the following
equations perpendicular, parallel, or
neither?
𝟐𝒙 + 𝟑𝒚 = 𝟗
𝟑
𝒙= 𝒚+𝟗
𝟐
𝟐𝒚 − 𝟑𝒙 = 𝟒
𝟐
𝒙=− 𝒚+𝟒
𝟑
Perpendicular
Congruent Triangles 4.2
Congruent Triangles: When two triangles are the
exact same size and shape they are said to be
congruent. Even though they have the same
shape and size, they may be positioned differently.
4.2 What corresponding parts of the two congruent
triangles are congruent?
z
a
y
b
x
c
𝒂𝒃 ≅ 𝒛𝒚
∠𝒃 ≅ ∠𝒚
𝒃𝒄 ≅ 𝒚𝒙
∠𝒂 ≅ ∠𝒛
𝒂𝒄 ≅ 𝒛𝒙
∠𝒄 ≅ ∠𝒙
∆𝒂𝒃𝒄 ≅ ∆𝒛𝒚𝒙
This slide demonstrates the concept of CPCTC. If two
triangles are congruent, then the corresponding parts
of those congruent triangles are congruent.
z
a
y
b
x
c
𝒂𝒃 ≅ 𝒛𝒚
∠𝒃 ≅ ∠𝒚
𝒃𝒄 ≅ 𝒚𝒙
∠𝒂 ≅ ∠𝒛
𝒂𝒄 ≅ 𝒛𝒙
∠𝒄 ≅ ∠𝒙
Congruent Triangles 4.2
Third Angle Theorem: If two angles of one triangle
are congruent to two angles of another triangle,
then the third angle must also be congruent.
𝟑𝟏°
𝟖𝟕°
𝟑𝟏°
𝟖𝟕°
𝟔𝟐°
𝟔𝟐°
4.2 Isosceles Triangles
Isosceles Triangle Conjecture:
An isosceles triangle has two
congruent angles.
The sides opposite the
congruent angles(legs) are
also congruent.
Vertex Angle
4.2 Isosceles Triangles
Converse of the Isosceles
Triangle Conjecture: A
triangle that has two
congruent angles must be
isosceles.
If a triangle has two
congruent sides, then it is
isosceles.
Vertex Angle
4.2 Isosceles Triangles
Vertex Angle: Angle between
the two congruent sides.
Legs: Congruent sides.
Base Angle
Base Angle
Base
4.3
By comparing only three parts of two different
triangles we will try to determine if the two triangles
are congruent. Basically, we will try to answer the
following question:
Is
≅
?
4.4
You have already seen that if two angles of one
triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
Which conjecture tells you this?
The Third Angle Conjecture
60 ̊
60 ̊
90 ̊
90 ̊
30 ̊
30 ̊
Just what is this congruence you speak of?
z
a
y
b
x
c
𝒂𝒃 ≅ 𝒛𝒚
S
∠𝒂𝒃𝒄 ≅ ∠𝒛𝒚𝒙
A
𝒃𝒄 ≅ 𝒚𝒙
S
∆𝒂𝒃𝒄 ≅ ∆𝒛𝒚𝒙
4.3
Side-Side-Side (SSS) Congruence Postulate: If three sides in
one triangle are congruent to corresponding sides in
another triangle, then the triangles are congruent.
d
s
e
t
w
How do we indicate their congruence?
∆𝒔𝒕𝒘 ≅ ∆𝒅𝒆𝒉
h
4.3
Side-Angle-Side (SAS) Congruence Postulate: If
corresponding sides and the included angle (angle
between the two congruent sides) are congruent in two
triangles, then the triangles are congruent.
w
z
t
y
x
How do we indicate their congruence?
∆𝒛𝒚𝒙 ≅ ∆𝒘𝒕𝒔
s
4.4
Angle-Side-Angle (ASA) Congruence Postulate: If
corresponding angles and the included side (the side
between the angles) are congruent in two triangles, then
the triangles are congruent.
x
b
y
c
a
How do we indicate their congruence?
∆𝒄𝒃𝒂 ≅ ∆𝒚𝒙𝒛
z
4.4
Side-Angle-Angle (SAA) Congruence Theorem: If
corresponding angles and a non-included side of two
triangles are congruent, then the triangles are congruent.
x
b
y
c
a
How do we indicate their congruence?
∆𝒄𝒃𝒂 ≅ ∆𝒚𝒙𝒛
z
4.6
Hypotenuse-Leg (HL) Congruence Theorem: If the
hypotenuse and a corresponding leg of two right triangles
are congruent, then the triangles are congruent.
A
B
D
C
E
How do we indicate their congruence?
∆𝑨𝑩𝑪 ≅ ∆𝑫𝑬𝑭
F
4.4
Side-Side-Angle (SSA) Congruence Conjecture: If
corresponding sides and a non-included angle of two
triangles are congruent, then the triangles are congruent.
x
b
y
c
a
not enough by itself
z
4.4
Angle-Angle-Angle (AAA) Congruence Conjecture: If two
triangles have all congruent angles, then the triangles are
congruent.
not enough by itself
Are the two triangles congruent? If
so, how do you know?
SAS
Are the two triangles congruent? If
so, how do you know?
SSS
Are the two triangles congruent? If
so, how do you know?
Not Enough Information
Are the two triangles congruent? If
so, how do you know?
SAS
Are the two triangles congruent? If
so, how do you know?
SSS
Are the two triangles congruent? If
so, how do you know?
AAS
Are the two triangles congruent? If
so, how do you know?
AAS
Are the two triangles congruent? If
so, how do you know?
ASA
Are the two triangles congruent? If
so, how do you know?
HLT
Are the two triangles congruent? If
so, how do you know?
SAA
Are the two triangles congruent? If
so, how do you know?
SAS
Practice
For the following slides, complete the two
column proofs.
Statements
Prove ∠𝑨 ≅ ∠𝑪
Reasons
1. 𝑨𝑫 ≅ 𝑫𝑪, 1. Given
∠𝑨𝑫𝑩 ≅ ∠𝑪𝑫𝑩
A
D
2. 𝑩𝑫 ≅ 𝑩𝑫 2. Reflexive
property
3. ∆𝑨𝑫𝑩 ≅ ∆𝑪𝑫𝑩 3. SAS
4. ∠𝑨 ≅ ∠𝑪 4. CPCTC
B
C
Statements
Prove: ∠𝑷𝑸𝑺 ≅ ∠𝑹𝑸𝑺
Reasons
1. 𝑷𝑸 ≅ 𝑸𝑹, 1. Given
𝑷𝑺 ≅ 𝑹𝑺
Q
2. 𝑸𝑺 ≅ 𝑸𝑺
2. Reflexive
property
3. ∆𝑷𝑸𝑺 ≅ ∆𝑹𝑸𝑺 3. SSS
4. ∠𝑷𝑸𝑺 ≅ ∠𝑹𝑸𝑺 4. CPCTC
P
R
S
Statements
Prove 𝑨𝑩 ≅ 𝑫𝑬
A
D
Reasons
1. 𝑨𝑫 ∥ 𝑩𝑬, 1. Given
∠𝑨 ≅ ∠𝑬
2. 𝑩𝑫 ≅ 𝑩𝑫 2. Reflexive
property
3. ∠𝑨𝑫𝑩 ≅ ∠𝑬𝑩𝑫 3. Alternate
Interior ∠s
Theorem
4. ∆𝑨𝑫𝑩 ≅ ∆𝑬𝑩𝑫 4. AAS
B
E
5. 𝑨𝑩 ≅ 𝑫𝑬
5. CPCTC
Theorem 4.6: Base Angles Theorem
If two sides of a triangle are congruent, then the
angles opposite them are congruent.
D
F
E
If 𝑫𝑭 ≅ 𝑬𝑭, 𝒕𝒉𝒆𝒏 ∠𝑫 ≅ ∠𝑬.
Theorem 4.7: Converse of the Base
Angles Theorem
If two angles of a triangle are congruent, then the
sides opposite them are congruent.
D
F
E
If ∠𝑫 ≅ ∠𝑬, 𝒕𝒉𝒆𝒏 𝑫𝑭 ≅ 𝑬𝑭.
Practice
What is the
length of 𝑩𝑪?
A
𝟏𝟗𝒄𝒎
C
B
𝑩𝑪 = 𝟏𝟗𝐜𝐦
Practice
What is the
degree measure
of ∠𝑫?
D
𝟔𝟒°
𝟒𝟓𝒄𝒎
𝟏𝟖𝟎 − 𝟓𝟐 = 𝟏𝟐𝟖
𝟓𝟐°
C
𝟏𝟐𝟖 ÷ 𝟐 = 𝟔𝟒
∠𝑫 = 𝟔𝟒°
𝟔𝟒°
E
𝟒𝟓𝒄𝒎
Corollaries
Corollary to Theorem 4.6
If a triangle is equilateral, then it must be equiangular.
Corollaries
Corollary to Theorem 4.7
If a triangle is equiangular, then it must be equilateral.
Practice
A
What is the
degree measure
of ∠𝑩?
𝟔𝟎°
𝟏𝟗𝒄𝒎
𝟏𝟗𝒄𝒎
𝟏𝟖𝟎 ÷ 𝟑 = 𝟔𝟎
∠𝑩 = 𝟔𝟎°
B
𝟔𝟎°
𝟔𝟎°
𝟏𝟗𝒄𝒎
C
5.1 Bisectors of a Triangle
Any point on the perpendicular bisector of a segment
must be equidistant from the endpoints of the segment.
A
B
C
𝑨𝑩 ≅ 𝑨𝑫
D
5.1 Bisectors of a Triangle
Concurrency of Perpendicular Bisectors of a Triangle: Perpendicular
bisectors of a triangle are all concurrent at a point called the
circumcenter. This point is equidistant from the all the vertices in
the triangle.
A
E
F
G
B
D
𝑨𝑮 = 𝑩𝑮 = 𝑪𝑮
C
5.3 Median
A median goes from a vertex in a triangle to the midpoint
of the opposite side.
A
E
F
G
B
D
C
5.3 Median
What are the medians in this triangle?
𝑭𝑪, 𝑩𝑬, and 𝑨𝑫 are medians.
A
E
F
G
B
D
C
5.3 Median
The three medians of a triangle are all concurrent at a
point called the centroid.
A
E
F
G
B
Point 𝑮 is the centroid.
D
C
Theorem 5.7 Concurrency of Medians of a Triangle
The centroid is located at two thirds of the distance from
each vertex to the midpoint to the opposite side.
If 𝑮 is the centroid ∆𝑨𝑩𝑪,
𝐭𝐡𝐞𝐧:
𝟐
𝑨𝑮 = 𝑨𝑫
𝟑
A
E
F
𝟐
𝑪𝑮 = 𝑪𝑭
𝟑
𝟐
𝑩𝑮 = 𝑩𝑬
𝟑
G
B
D
C
𝑮 is the centroid of ∆𝑨𝑩𝑪.
A
1. If 𝑨𝑮 = 𝟐𝟒𝒄𝒎, what is the
Length of 𝑨𝑫 ?
𝑨𝑫 = 𝟑𝟔𝒄𝒎
2. If 𝑭𝑮 = 𝟖𝒄𝒎, what is the
Length of 𝑭𝑪 ?
𝑭𝑪 = 𝟐𝟒𝒄𝒎
3. If 𝑩𝑬 = 𝟒𝟓𝒄𝒎, what is
the length of 𝑩𝑮 ?
𝑩𝑮 = 𝟑𝟎𝒄𝒎
E
F
G
B
D
C
5.3 Altitude
An altitude in a triangle goes from a vertex and creates a
right angle (is perpendicular with) the opposite side. An
altitude can lie in, on, or outside the triangle.
X
B
W
Y
Z
R
S
A
T
D
C
5.3 Altitude
All of the altitudes in a triangle are concurrent at a single
point called the orthocenter.
5.4
Midsegment: segment connecting midpoints of
two sides of a triangle.
A
𝑮𝑩, 𝑩𝑫, and 𝑫𝑮 are
all midsegments.
G
E
B
D
C
5.4
𝟏
𝟐
Midsegment Theorem: a midsegment is the
length of the side of the triangle it is parallel to.
A
G
𝟏
𝑮𝑩 = 𝑬𝑪
𝟐
𝟏
𝑩𝑫 = 𝑨𝑬
𝟐
B
𝟏
𝑮𝑫 = 𝑨𝑪
𝟐
E
D
C
Practice
𝑭𝑩, 𝑩𝑫, and 𝑫𝑭 are all midsegments.
Find the lengths of segments 𝑨𝑪, 𝑩𝑫, and 𝑪𝑬.
B
A
C
4
𝑨𝑪 = 𝟏𝟎
11 F
5
D
𝑩𝑫 = 𝟓. 𝟓
𝑪𝑬 = 𝟖
E
Practice
𝑮𝑱 = Midsegment
G
R
T
J
𝑩𝑫 = Altitude
B
S
𝑿𝑾 = Median
X
A
D
Z
W
C
Y
5.5
Theorem 5.10 Side-Angle Inequality Theorem: The longest
side in a triangle is opposite the largest angle. The shortest
side in a triangle is opposite the smallest angle. The side
with length in between the lengths of the other sides is
opposite the angle that has degree measure in between the
other angles.
𝟓𝒄𝒎
𝟑𝟖°
𝟓𝟔°
𝟑𝒄𝒎
𝟒𝒄𝒎
𝟖𝟔°
Side-Angle Inequality
B
𝟏𝟎𝟑°
𝟒𝟖°
𝟐𝟗°
A
C
List the sides in order
from longest to shortest.
𝑨𝑪 > 𝑨𝑩 > 𝑩𝑪
5.5
Theorem 5.13 Triangle Inequality: The sum of the length
of two sides of a triangle must be greater than the length
of the third side.
B
8cm
10cm
A
C
17cm
𝟏𝟎 + 𝟖 > 𝟏𝟕
𝟏𝟎 + 𝟏𝟕 > 𝟖
∆𝑨𝑩𝑪 could be a triangle.
𝟖 + 𝟏𝟕 > 𝟏𝟎
5.5
Theorem 5.13 Triangle Inequality: The sum of the length
of two sides of a triangle must be greater than the length
of the third side.
B
8cm
10cm
A
C
20cm
∆𝑨𝑩𝑪 could not be a triangle.
𝟏𝟎 + 𝟖 > 𝟐𝟎
Practice
Determine whether it is possible to draw a triangle with
the sides of the given measurements.
 8cm, 11cm , 18.5cm No
 7cm, 17cm , 10.5cm Yes
 9cm, 6cm , 19cm No
How do you know?
5.6
Hinge Theorem
 Sometimes a picture is worth a thousand confusing
math terms.
 Which side do you predict will be longer 𝑨𝑩 or 𝑫𝑬 ?
C
𝑨𝑩 < 𝑫𝑬
𝟗𝟕°
A
B
F
𝟏𝟏𝟎°
E
D
Practice
Fill in the blank with a “less than”, “greater than”, or
“equal to” symbol.
𝑪𝑩 > 𝑭𝑬
C
F
B
𝟖𝟐°
E
A
𝟕𝟏°
D
Practice
For the following slides, complete the two
column or flow chart proofs.
Practice
Prove that the two triangles are congruent
Statements
a
Reasons
b 1. ∠𝒂𝒄𝒃 ≅ ∠𝒅𝒃𝒄 1. Given
2. 𝒃𝒄 ≅ 𝒃𝒄
2. Reflexive
property
3. ∠𝒂𝒃𝒄 ≅ ∠𝒅𝒄𝒃 3. Alternate
Interior ∠s
Theorem
4. ∆𝒂𝒃𝒄 ≅ ∆𝒅𝒄𝒃 4. ASA
c
d
Prove that 𝒄𝒓 ≅ 𝒘𝒐.
Statements
w
c
Reasons
1. Given
1. 𝒄𝒏 ≅ 𝒘𝒏,
𝒓𝒏 ≅ 𝒐𝒏
2. ∠𝒄𝒏𝒓 ≅ ∠𝒘𝒏𝒓 2. Vertical
Angles Theorem
n
3. ∆𝒄𝒏𝒓 ≅ ∆𝒘𝒏𝒐 3. SAS
r
4. 𝒄𝒓 ≅ 𝒘𝒐
o
4. CPCTC
Given: 𝑩𝑪 ≅ 𝑪𝑫, 𝑨𝑩 ll 𝑫𝑬
Prove: 𝑨𝑪 ≅ 𝑪𝑬
A
D
C
B
E
Flow Chart
a
d
Prove ∠𝒂 ≅ ∠𝒅
b
c
e
f
𝒂𝒄 ≅ 𝒅𝒇
Given
∠𝒄 ≅ ∠𝒇
∆𝒂𝒃𝒄 ≅ ∆𝒅𝒆𝒇
∠𝒂 ≅ ∠𝒅
Given
AAS
CPCTC
∠𝒃 ≅ ∠𝒆
Right ∠ Congruence Theorem
a
𝒂𝒃 is an altitude of ∆𝒂𝒄𝒅
Prove ∠𝒄 ≅ ∠𝒅
c
b
d
𝒄𝒃 ≅ 𝒃𝒅
Given
∠𝒂𝒃𝒄 ≅ ∠𝒂𝒃𝒅
∆𝒂𝒃𝒄 ≅ ∆𝒂𝒃𝒅
∠𝒄 ≅ ∠𝒅
Definition of an altitude
SAS
CPCTC
𝒂𝒃 ≅ 𝒂𝒃
Reflexive Property
Fill in the blanks. Prove ∠𝑩 ≅ ∠𝑫.
Given: C is the midpoint of 𝑩𝑫, 𝑨𝑩 ≅ 𝑨𝑫
A
Statements
1. 𝑨𝑩 ≅ 𝑨𝑫
1. Given
2. 𝑨𝑪 ≅ 𝑨𝑪
2. Reflexive
Property
3. Definition
of a
midpoint
3. 𝑩𝑪 ≅ 𝑪𝑫
4. ∆𝑨𝑩𝑪 ≅
∆𝑨𝑫𝑪
B
C
Reasons
D 5. ∠𝑩 ≅ ∠𝑫
4. SSS
5. CPCTC
Prove angle B is congruent to angle D.
𝑨𝑪 is a median of triangle ABD.
A
Statements
B
C
D
Reasons
Given: 𝑿𝒀 ≅ 𝑿𝑾, 𝒀𝑾 ⊥ 𝑿𝒁
Prove ∠𝒀 ≅ ∠𝑾
X
Statements
1. 𝑿𝒀 ≅ 𝑿𝑾,
𝒀𝑾 ⊥ 𝑿𝒁
Reasons
1. Given
2. ∠𝑿𝒁𝒀 ≅ ∠𝑿𝒁𝑾 2. Definition of
⊥ lines
3. 𝑿𝒁 ≅ 𝑿𝒁
3. Reflexive
Property
4. ∆𝑿𝒁𝒀 ≅ ∆𝑿𝒁𝑾 4. HLT
5. ∠𝒀 ≅ ∠𝑾 5. CPCTC
Y
Z
W
Practice
𝒚+𝟗
𝒙−𝟖
23
Practice
x
62 ̊
y
x
110 ̊
44 ̊
135 ̊
77 ̊
z