1/15/20144-2: measuring angles in triangles expectations: g1.2.1: prove that the sum of the angle...
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04/10/23 4-2: Measuring Angles in Triangles
4-2: Measuring Angles in Triangles
Expectations:
G1.2.1: Prove that the sum of the angle measures of a triangle is 180 and that the measure of an exterior angle of a triangle is the sum of the measures of the 2 remote interior angles.
G1.2.2: Solve problems involving angle measure and side length of triangles.
04/10/23 4-2: Measuring Angles in Triangles
Linear Triples
∠A, B and C form a linear triple iff A is ∠ ∠ ∠adjacent to B, B is adjacent to C, ∠ ∠ ∠
A and C have no points in common ∠ ∠(other than the vertex) and the sum of their measures is 180.
∠A ∠B ∠C
∠A, ∠B and ∠C form a linear triple.
04/10/23 4-2: Measuring Angles in Triangles
Justify the sum of the measures of A, B and C.∠ ∠ ∠
AB
C
04/10/23 4-2: Measuring Angles in Triangles
Triangle Sum Theorem:
The sum of the measures of the angles of a triangle is _______°.
04/10/23 4-2: Measuring Angles in Triangles
Determine the value of x.
A
C
B58 x - 8
x
In triangle ABC below, the measure of A is ∠20° and the measure of B is 3 times larger ∠than the measure of C. What is the ∠measure of angle B?
A.40°
B.60°
C.80°
D.120°
E.160°
04/10/23 4-2: Measuring Angles in Triangles
A
B
C
04/10/23 4-2: Measuring Angles in Triangles
Calculate the measures of each angle below.
7x+3 5x+9
8x-12
A
C
B
04/10/23 4-2: Measuring Angles in TrianglesA
C
B
F
ED
If ∠A≅∠D and ∠C≅∠F, what may you conclude about ∠B and ∠E?
Can you justify your conjecture?
04/10/23 4-2: Measuring Angles in Triangles
04/10/23 4-2: Measuring Angles in Triangles
Third Angle Congruence Theorem:
If two angles of one triangle are _____________ to 2 angles of a second triangle, then the third angles are also _____________ .
04/10/23 4-2: Measuring Angles in Triangles
Exterior Angles
An angle is an exterior angle of a triangle iff it is the union of one side of the triangle and the extension of the other side that shares the same vertex as the first side.
04/10/23 4-2: Measuring Angles in Triangles
Exterior Angles
A B
C
D
∠CBD is an exterior angle for ∆ABC
04/10/23 4-2: Measuring Angles in Triangles
Exterior Angles
A B
C
E
∠ABE is an exterior angle for ∆ABC
04/10/23 4-2: Measuring Angles in Triangles
Exterior Angles
Because there are 2 exterior angles at each vertex, every triangle has a total of 6 exterior angles.
04/10/23 4-2: Measuring Angles in Triangles
Remote Interior Angles:
A remote interior angle of a triangle is one of the 2 interior angles not adjacent to an exterior angle.
Remote interior angles change depending on which interior angle is being discussed.
04/10/23 4-2: Measuring Angles in Triangles
A C
B
D
Remote Interior Angles:∠ __ and ∠__ are remote interior angles for exterior ∠BCD.
04/10/23 4-2: Measuring Angles in Triangles
A C
B
E
Remote Interior Angles:∠__ and ∠__ are remote interior
angles for exterior ∠BAE.
04/10/23 4-2: Measuring Angles in Triangles
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is the sum of the measures of
04/10/23 4-2: Measuring Angles in Triangles
Exterior Angle Theorem: A Proof
A C
B
D
Given: ΔABC with exterior angle∠BCDProve: m∠BCD = m∠A + m∠B
04/10/23 4-2: Measuring Angles in Triangles
Given: ΔABC with exterior angle ∠BCDProve: m∠BCD = m∠A + m∠B
AC
B
D
04/10/23 4-2: Measuring Angles in Triangles
What is the measure of ∠A?
124°
A
52°
04/10/23 4-2: Measuring Angles in Triangles
A corollary is a theorem that is easily proven by another theorem.
Corollary
04/10/23 4-2: Measuring Angles in Triangles
Acute Angles Corollary
The acute angles of a right triangle are ____________.
A
C B
m∠A + m∠B =
04/10/23 4-2: Measuring Angles in Triangles
Right Angle Corollary
There can be at most ___ right angle in a triangle.
Obtuse Angle Corollary
There can be at most ___ obtuse angle in a triangle.
04/10/23 4-2: Measuring Angles in Triangles
In triangle ABC, m A is 16 more than m B ∠ ∠and m C is 29 more than m B.∠ ∠
Determine the measure of each angle.
04/10/23 4-2: Measuring Angles in Triangles
04/10/23 4-2: Measuring Angles in Triangles
If AB is perpendicular to BC, determine the measures of all 8 numbered angles.
04/10/23 4-2: Measuring Angles in Triangles
Assignment:
pages 193-195,
# 15-19 (odds), 36, 39, 42-44 (all)