Unit 4 Review

Warm Up Grab a gold square from the front of the room and fold it into four boxes

TRIANGLE Definition: A Triangle is a three-sided polygon Characteristics: Has three sides and three angles Real life examples:

TRIANGLE ANGLE SUM THEOREM The sum of the measures of the angles of a triangle is equal to 180.

EXAMPLE 1

EXAMPLE 2

CLASSIFYING TRIANGLES We will classify using SIDE LENGTHS and ANGLES. Triangles can fit into more than one category

CLASSIFYING BY SIDE LENGTHS Look at the SIDES of the triangle

SIDE LENGTHS SCALENE Triangle- Triangle with all sides different lengths

SIDE LENGTHS Equilateral Triangle- triangle with all sides congruent

SIDE LENGTHS ISOSCELES Triangle- Triangle with at least two congruent sides

CLASSIFYING BY ANGLES Look at the ANGLES of the triangle

ANGLE MEASUREMENT OBTUSE Triangle- triangle with an obtuse angle Obtuse definition?

ANGLE MEASUREMENT ACUTE Triangle- Triangle where all angles are acute angles Acute definition?

ANGLE MEASUREMENT RIGHT Triangle- A triangle with one right angle

EQUILATERAL/ EQUIANGULAR Equilateral triangles are ALWAYS equiangular. Equiangular triangles are ALWAYS equilateral.

EXAMPLE 1: (PAGE 14)  RED 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 ED

EXAMPLE 2: (PAGE 14)  UNC is isosceles, UN = 3x – 2, NC = 2x + 1, and UC = 5x – 2. Find x and the measure of each side of the triangle. U N C

ISOSCELES TRIANGLES s in which 2 or more sides are

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

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

CONVERSE OF THE ISOSCELES TRIANGLE THEOREM If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

EXAMPLE 1 (PAGE 20) Find x

EXAMPLE 2 Find x.

EXAMPLE 3 Find x.

EXAMPLE 4 Find x.

EXAMPLE 5 Find x.

+ Exterior Angle Theorem for Triangles The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

+ Example 1 Find m  1.

+ Example 2 Find x.

Points of Concurrency NOT CONCURRENT!

Altitude/Orthocenter of a Triangle  Definition: An Altitude of a triangle is a segment that extends from vertex of a triangle and is perpendicular to the opposite side.

Looking at Altitudes of a Triangle  With one of the triangles given to you, use a ruler to draw the three altitudes.  Are the altitudes concurrent?  Yes!  Definition: The Orthocenter is the point of intersection of the three altitudes of a triangle

Medians/Centroid of a Triangle  Definition: A Median of a triangle is a segment that connects a vertex to the median of the opposite side.

Looking at Medians of a Triangle  With one of the triangles given to you, connect two vertices and crease the middle. This is the midpoint  Repeat for the other two sides.  Use your ruler to trace a line from each midpoint to the opposite vertex.  Are the medians concurrent?  Yes!  Definition: The Centroid is the point of intersection of the three medians of a triangle

3 6 More on Centroids  The centroid of the triangle divides each median into two parts.  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.  In other words, the two parts have a ratio of 2:1.

Guided Practice – Page 36

Review – Angle Bisector  Definition: A line that cuts an angle into two equal parts.

Looking at Angle Bisectors of a Triangle  With one of the triangles given to you, connect two edges. Crease all the way down This is an angle bisector.  Repeat for the other two angles.  Use your ruler to trace the lines.  Are the angle bisectors concurrent?  Yes!  Definition: The Incenter is the point of intersection of the three angle bisectors of a triangle

Review – Perpendicular Bisector  Definition: A perpendicular line that cuts a segment into two equal parts.

Looking at Perpendicular Bisectors of a Triangle  With one of the triangles given to you, connect two vertices. Crease all the way down This is an perpendicular bisector.  Repeat for the other two sides.  Use your ruler to trace the lines.  Are the perpendicular bisectors concurrent?  Yes!  Definition: The Circumcenter is the point of intersection of the three perpendicular bisectors of a triangle