Download TRIANGLE PROPERTIES • Interior angles of a triangle add up to

TRIANGLE PROPERTIES
• Interior angles of a triangle add up to 180°.
This property can be deductively proven, so it is a theorem.
PQ  AB
example: Given:
P
C
Q
Prove: ∠CAB + ∠ACB + ∠CBA = 180°
A
statement
reason
PQ  AB
given
∠ACP = ∠CAB
alternate interior angles
∠BCQ = ∠CBA
alternate interior angles
B
∠ACP + ∠ACB + ∠BCQ = 180° angles on a line add up to 180°
∠CAB + ∠ACB + ∠CBA = 180° substituting equal angles
Q.E.D.
An exterior angle (external angle) of a triangle is the angle between one side of a
triangle and the extension of an adjacent side.
C
exterior angle
B
A
D
Use the second triangle (above right) to draw other exterior angles.
example: Prove the conjecture, "an exterior angle at a vertex of a triangle is
equal to the sum of the other two interior angles in the triangle."
C
Given: ∆ABC with exterior angle ∠CAD
Prove: ∠CAD = ∠ABC + ∠ACB
B
D
A
Statement
Reason
∠CAB + ∠CAD = 180°
angles on a line are supplementary
∠CAD = 180° − ∠CAB
subtraction property
∠CAB + ∠B + ∠C = 180°
sum of angles in a triangle
∠B + ∠C = 180° − ∠CAB
subtraction property
∠CAD = ∠B + ∠C
transitive property
Q.E.D.
Theorem: An exterior angle at a vertex of a triangle is equal to the sum of the
other two interior angles in the triangle.
exercise: Find the measure of each unknown angle.
46°
67°
1
2
20°
4
110°
3
∠1 =
∠3 =
∠2 =
∠4 =
∠5 =
9 38°
8
6
7
130°
∠6 =
∠8 =
∠7 =
∠9 =
5
USING TRIANGLE PROPERTIES
• Interior angles of a triangle add up to 180°.
• If a triangle is isosceles, then the angles opposite the equal sides are equal.
• If a triangle is equilateral, then all the angles are equal to each other and 60°.
• An exterior angle at a vertex of a triangle is equal to the sum of the other two
interior angles in the triangle.
exercise: Given: ∆ ABC is equilateral
A
∠ADC = 35°
Prove: ∠CAD = 25°
B
statement
∆ ABC is equilateral
C
D
reason
given
property of equilateral triangles
∠ACD =
°
∠ADC = 35°
angles in a triangle add up to 180°
∠CAD + 120° + 35° = 180°
substitution
∠CAD = 25°
Q.E.D.
exercise: Determine the interior angles of the triangle.
(x + 5)°
x°
(6x − 25)°
Answer:
exercise: Determine the perimeter of the isosceles triangle.
(2x + 17) cm
(3x − 20) cm
(5x − 25) cm
Answer:
exercise: Determine the interior angles in the triangle.
Answer: