Sec 4.1 Apply Triangle Sum Properties
VOCABULARY
Triangle
______
Interior angles: angles inside the figure.
Ex: angles 4, 6, and 7.
Exterior angles: angles created by extending
the sides of a figure,
lie outside the figure.
Ex: angles 1, 2, 3, 5, 8, 9, 10, 11, 12.
Corollary to a theorem
______
______
CLASSIFYING TRIANGLES BY SIDES
Scalene Triangle / Isosceles Triangle / Equilateral Triangle____ congruent sides / At least ____ congruent sides / ____ congruent sides
CLASSIFYING TRIANGLES BY ANGLES
Acute Triangle / Right Triangle / Obtuse Triangle / Equiangular Triangle____ acute angles / ____ right angle / ____ obtuse angles / ____ congruent angles
Example 1 Classify ∆RST by its sides. Determine if the triangle is a right triangle.
Step 1 Use the distance formula to find the side lengths. RS = __8.1__ ST = ______RT = ______
Step 2 Check for right angles.
The slope of is ______. The slope of is ______.
The product of the slopes is ______, so ^ and ÐRTS is a ______angle.
Therefore, ∆RST is a ______triangle.
THEOREM 4.1: TRIANGLE SUM THEOREM
mÐA + mÐB + mÐC = ______
The sum of the measures of the interior angles of a triangle is ______.
THEOREM 4.2: EXTERIOR ANGLE THEOREM
Angles A, B and C are interior angles. A and B are called REMOTE INTERIOR angles because they are far away from the exterior angle at C.
mÐ1 = mÐ_ __ + mÐ__ _
The measure of an exterior angle of a triangle is equal to the sum of the measures
of the two remote interior angles.
Example 2
Use the diagram at the right to find the measure of ÐDCB, ÐA and Ð1 .
Solve 3x+16 = 52 + 2x
x = 36
so mÐA=2(36) = 72,
mÐ1 = 180-(52+72) = 56.
mÐDCB= 3(36)+16 = 124.
To check your answers:
The exterior angle = the sum of the 2 remote interior angles. 124 = (52+72) smile!
Example 3
Use the diagram at the right to find the measure of ÐDCB.
Solution
Step 1 Write and solve an equation to find the value of x.
3x - 9 = x + 73 / Exterior Angle Theoremx = _____ / Solve for x.
Step 2 Substitute x = ____ in 3x - 9 to find mÐDCB.
3x - 9 = 3 · ____ - 9 = _____.
The measure of ÐDCB is _____.
COROLLARY TO THE TRIANGLE SUM THEOREM
mÐA + mÐB = _____
The acute angles of a right triangle are ______.