Angles of Triangles
I can classify triangles by sides and angles and
find interior and exterior angle measures.
Angles of Triangles
Vocabulary (page 125 in Student Journal)
corollary to a theorem: a statement that can be
proved easily using the theorem
Angles of Triangles
Core Concepts (page 126 in Student Journal)
Triangle Sum Theorem (Theorem 5.1)
The sum of the measures of the interior angles of a
triangle is 180 degrees.
Exterior Angle Theorem (Theorem 5.2)
The measure of an exterior angle of a triangle is
equal to the sum of the measures of the 2
nonadjacent interior angles.
Angles of Triangles
Corollary to the Triangle Sum Theorem
(Corollary 5.1)
The acute angles of a right triangle are
complementary.
Angles of Triangles
Examples (space on pages 125 and 126 in
Student Journal)
a) Classify triangle ABC by its sides and its
angles.
Angles of Triangles
Solutions
a) scalene and acute
Angles of Triangles
b) Find the measure of angle PQS.
Angles of Triangles
Solution
b) 145 degrees
Angles of Triangles
c) The measure of 1 acute angle of a right
triangle is 1.5 times the measure of the other
acute angle. Find the measure of each acute
angle.
Angles of Triangles
Solution
c) 36 and 54 degrees
Congruent Polygons
I can use rigid motions to map 1 triangle to
another triangle.
Congruent Polygons
Vocabulary (page 130 in Student Journal)
corresponding parts: each angle and side from
the preimage matches to an angle and side of
the image
Corresponding parts of congruent triangles are
congruent. (CPCTC)
Congruent Polygons
Core Concepts (page 130 in Student Journal)
Properties of Triangle Congruence (Theorem 5.3)
The reflexive, symmetric, and transitive properties apply
to triangle congruence.
Third Angles Theorem (Theorem 5.4)
If 2 angles in 1 triangle are congruent to 2 angles in
another triangle, then the 3rd angles are congruent.
Congruent Polygons
Examples (space on page 130 in Student Journal)
a) Write a congruence statement for the triangles. Then
identify all pairs of congruent corresponding parts.
Congruent Polygons
Solutions
a)
Congruent Polygons
In the diagram, DEFG is congruent to QMNP.
b) Find the value of x and y.
Congruent Polygons
Solution
b) x = 10, y = 27
Congruent Polygons
c) Find the measure of angle P.
Congruent Polygons
Solution
c) 38 degrees
Congruent Polygons
d) Prove that triangle WXY is congruent to triangle ZVY.
Congruent Polygons
Solution
d)
Proving Triangle Congruence
by SAS
I can prove triangles congruent by the SideAngle-Side Congruence Theorem.
Proving Triangle Congruence
by SAS
Core Concepts (page 135 in Student Journal)
Side-Angle-Side (SAS) Congruence Theorem
(Theorem 5.5)
If 2 sides and the included angle of 1 triangle are
congruent to 2 sides and the included angle of
another triangle, then the 2 triangles are
congruent.
Proving Triangle Congruence
by SAS
Examples (space on page 135 in Student Journal)
a) Prove triangle ABC is congruent to triangle
DBC, given that B is the midpoint of segment
AD.
Proving Triangle Congruence
by SAS
Solution
a)
Proving Triangle Congruence
by SAS
b) What can you conclude about triangle PTS and
triangle RTQ from the diagram?
Proving Triangle Congruence
by SAS
Solution
b) Triangle PTS and triangle RTQ are congruent
by SAS.
Proving Triangle Congruence
by SAS
c) Can you can conclude triangle IPA and
triangle IPR are congruent? If so, what other
information is needed that is not marked in the
diagram?
Proving Triangle Congruence
by SAS
Solution
c) Triangle IPA is congruent to triangle IPR by
SAS., because segment IP is congruent to itself
using the Reflexive Property of Congruence.
Equilateral and Isosceles
Triangles
I can make conjectures about side lengths and
angle measures of isosceles triangles.
Equilateral and Isosceles
Triangles
Vocabulary (page 140 in Student Journal)
legs: the congruent sides of an isosceles triangle
vertex angle: the angle formed by the legs in an
isosceles triangle
Equilateral and Isosceles
Triangles
base: the third noncongruent side of an isosceles
triangle
base angles: the other 2 angles in an isosceles
Equilateral and Isosceles
Triangles
Core Concepts (pages 140 and 141 in Student
Journal)
Base Angles Theorem (Theorem 5.6)
If 2 sides (legs) of a triangle are congruent, then
the angles (base angles) opposite those sides
are congruent.
Equilateral and Isosceles
Triangles
Converse of the Base Angles Theorem
(Theorem 5.7)
If 2 angles (base angles) of a triangle are
congruent, then the sides (legs) opposite those
angles are congruent.
Equilateral and Isosceles
Triangles
Corollary to the Base Angles Theorem
(Corollary 5.2)
If a triangle is equilateral, then it is equiangular.
Corollary to the Converse of the Base Angles
Theorem (Corollary 5.3)
If a triangle is equiangular, then it is equilateral.
Equilateral and Isosceles
Triangles
Examples (space on pages 140 and 141 in
Student Journal)
a) Name 2 congruent sides in the figure.
Equilateral and Isosceles
Triangles
Solution
a) segment AC is congruent to segment BC
Equilateral and Isosceles
Triangles
b) Find the value of x.
Equilateral and Isosceles
Triangles
Solution
b) x = 30
Equilateral and Isosceles
Triangles
c) Find the value of x and y.
Equilateral and Isosceles
Triangles
Solution
c) x = 15, y = 7
Equilateral and Isosceles
Triangles
In the diagram, segment PT is congruent to
segment ST and segment PQ is congruent to
segment SR.
d) Prove triangle PQT is congruent to triangle
SRT.
e) What kind of triangle is triangle QRT? Explain.
Equilateral and Isosceles
Triangles
Solutions
d)
e) Isosceles, because triangle PQT is congruent
to triangle SRT, so segment TQ is congruent to
segment TR by CPCTC.
Proving Triangle Congruence by
SSS
I can prove triangles congruent by Side-SideSide Congruence Theorem and Hypotenuse-Leg
Theorem.
Proving Triangle Congruence by
SSS
Core Concepts (page 145 in Student Journal)
Side-Side-Side (SSS) Congruence Theorem
(Theorem 5.8)
If 3 sides of 1 triangle are congruent to 3 sides of
a second triangle, then the 2 triangles are
congruent.
Proving Triangle Congruence by
SSS
Hypotenuse-Leg (HL) Congruence Theorem
(Theorem 5.9)
If the hypotenuse and the leg of a right triangle
are congruent to the hypotenuse and the leg of a
second right triangle, then the 2 triangles are
congruent.
Proving Triangle Congruence by
SSS
Examples (space on page 145 in Student Journal)
a)
Proving Triangle Congruence by
SSS
Solution
a)
Proving Triangle Congruence by
SSS
b)
Proving Triangle Congruence by
SSS
Solution
b)
Proving Triangle Congruence
by ASA and AAS
I can prove triangles congruent by the AngleSide-Angle and Angle-Angle-Side Congruence
Theorems.
Proving Triangle Congruence
by ASA and AAS
Core Concepts (page 150 in Student Journal)
Angle-Side-Angle (ASA) Congruence Theorem
(Theorem 5.10)
If 2 angles and the included side of 1 triangle are
congruent to 2 angles and the included side of
another triangle, then the 2 triangles are
congruent.
Triangle Congruence by ASA
and AAS
Angle-Angle-Side (AAS) Congruence Theorem
(Theorem 5.11)
If 2 angles and a nonincluded side of 1 triangle
are congruent to 2 angles and the
corresponding nonincluded side of another
triangle, then the triangles are congruent.
Triangle Congruence by ASA
and AAS
Examples (space on page 150 in Student Journal)
Can you prove the triangles in the diagram
congruent? Explain
a)
b)
c)
Triangle Congruence by ASA
and AAS
Solutions
a) yes, AAS
b) no, not enough information
c) yes, ASA
Triangle Congruence by ASA
and AAS
d)
Triangle Congruence by ASA
and AAS
Solution
d)
Using Congruent Triangles
I can use congruent triangles to make an indirect
measurement.
Using Congruent Triangles
Examples (space on page 155 in Student Journal)
a) Find the distance across the pond in the
diagram. Explain your solution.
Using Congruent Triangles
Solutions
a)
Using Congruent Triangles
b) Prove triangle ADE is congruent to triangle
ABE.
Using Congruent Triangles
Solution
b)
Coordinate Proofs
I can use a coordinate plane to write a proof.
Coordinate Proofs
Vocabulary (page 160 in Student Journal)
coordinate proof: placing figures in the
coordinate plane and showing a relationship
that exists between those figures using
midpoints, distances, slopes, transformations,
etc.
Coordinate Proofs
Examples (space on page 160 in Student Journal)
a) Prove ray SO bisects angle PSR.
Coordinate Proofs
Solution
a)
Coordinate Proofs
b) Prove angle TOU is congruent to angle VUO.
Coordinate Proofs
Solution
b)
Coordinate Proofs
c) Prove MNPQ has 4 congruent sides.
Coordinate Proofs
Solution
c)