Warm Up. Grab a gold square from the front of the room and fold it into four boxes
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1 Unit 4 Review
2 Warm Up Grab a gold square from the front of the room and fold it into four boxes
3 TRIANGLE Definition: A Triangle is a three-sided polygon Characteristics: Has three sides and three angles Real life examples:
4 TRIANGLE ANGLE SUM THEOREM The sum of the measures of the angles of a triangle is equal to 180.
5 EXAMPLE 1
6 EXAMPLE 2
7 CLASSIFYING TRIANGLES We will classify using SIDE LENGTHS and ANGLES. Triangles can fit into more than one category
8 CLASSIFYING BY SIDE LENGTHS Look at the SIDES of the triangle
9 SIDE LENGTHS SCALENE Triangle- Triangle with all sides different lengths
10 SIDE LENGTHS Equilateral Triangle- triangle with all sides congruent
11 SIDE LENGTHS ISOSCELES Triangle- Triangle with at least two congruent sides
12 CLASSIFYING BY ANGLES Look at the ANGLES of the triangle
13 ANGLE MEASUREMENT OBTUSE Triangle- triangle with an obtuse angle Obtuse definition?
14 ANGLE MEASUREMENT ACUTE Triangle- Triangle where all angles are acute angles Acute definition?
15 ANGLE MEASUREMENT RIGHT Triangle- A triangle with one right angle
16 EQUILATERAL/ EQUIANGULAR Equilateral triangles are ALWAYS equiangular. Equiangular triangles are ALWAYS equilateral.
17 EXAMPLE 1: (PAGE 14) ΔRED is equilateral with RE = x + 5, ED = 3x 9, and RD = 2x 2. Find x and the measure of each side of the triangle. R E D
18 EXAMPLE 2: (PAGE 14) ΔUNC is isosceles, UN = 3x 2, NC = 2x + 1, and UC = 5x 2. Find x and the measure of each side of the triangle. N U C
19 ISOSCELES TRIANGLES s in which 2 or more sides are
20 DEFINING LABELS Leg- the two congruent sides Base- the third non- congruent side Base Angle- the two angles created by each leg meeting the base Vertex Angle- the angle created by the two legs
21 ISOSCELES TRIANGLE THEOREM If two sides of a triangle are congruent, then the angles opposite those sides are congruent. SO. Our BASE ANGLES are ALWAYS CONGRUENT! Base angles are congruent m<n = m<c
22 CONVERSE OF THE ISOSCELES TRIANGLE THEOREM If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
23 EXAMPLE 1 (PAGE 20) Find x
24 EXAMPLE 2 Find x.
25 EXAMPLE 3 Find x.
26 EXAMPLE 4 Find x.
27 EXAMPLE 5 Find x.
28 + Exterior Angle Theorem for Triangles n The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
29 + Example 1 n Find m 1.
30 + Example 2 n Find x.
31 Points of Concurrency NOT CONCURRENT!
32 Altitude/Orthocenter of a Triangle ì DefiniDon: An Al#tude of a triangle is a segment that extends from vertex of a triangle and is perpendicular to the opposite side.
33 Looking at Altitudes of a Triangle ì With one of the triangles given to you, use a ruler to draw the three aldtudes. ì Are the aldtudes concurrent? ì Yes! ì DefiniDon: The Orthocenter is the point of intersecdon of the three aldtudes of a triangle
34 Medians/Centroid of a Triangle ì DefiniDon: A Median of a triangle is a segment that connects a vertex to the median of the opposite side.
35 Looking at Medians of a Triangle ì With one of the triangles given to you, connect two verdces and crease the middle. This is the midpoint ì Repeat for the other two sides. ì Use your ruler to trace a line from each midpoint to the opposite vertex. ì Are the medians concurrent? ì Yes! ì DefiniDon: The Centroid is the point of intersecdon of the three medians of a triangle
36 More on Centroids ì The centroid of the triangle divides each median into two parts. ì ì ì The distance from the centroid to the vertex is 2/3 the median The distance from the centroid to the side is 1/3 the median. The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint. ì In other words, the two parts have a rado of 2:1. 6 3
37 Guided Practice Page 36
38 Guided Practice Page 36
39 Guided Practice Page 36
40 Guided Practice Page 36
41 Guided Practice Page 36
42 Review Angle Bisector ì DefiniDon: A line that cuts an angle into two equal parts.
43 Looking at Angle Bisectors of a Triangle ì With one of the triangles given to you, connect two edges. Crease all the way down This is an angle bisector. ì Repeat for the other two angles. ì Use your ruler to trace the lines. ì Are the angle bisectors concurrent? ì Yes! ì DefiniDon: The Incenter is the point of intersecdon of the three angle bisectors of a triangle
44 Review Perpendicular Bisector ì DefiniDon: A perpendicular line that cuts a segment into two equal parts.
45 Looking at Perpendicular Bisectors of a ì With one of the triangles given to you, connect two verdces. Crease all the way down This is an perpendicular bisector. ì Repeat for the other two sides. ì Use your ruler to trace the lines. ì Are the perpendicular bisectors concurrent? ì Yes! Triangle ì DefiniDon: The Circumcenter is the point of intersecdon of the three perpendicular bisectors of a triangle
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