1. Section 1.5
Triangles & Angles
10th Grade Geometry
Mr. ONeills Class

2. Standards & Objectives
Standard :
Students will learn and apply geometric concepts.
Objectives:
Classify triangles by their sides and angles.
Find angle measures in triangles
DEFINITION:
A triangle is a figure formed by three segments joining three non-collinear points.
3. Names of Triangles
Triangles can be classified by the sides or by the angle
Scaleneno congruent sides
Equilateral3 congruent sides
Isosceles2 congruent sides
4. Equilateral Triangle
3 congruent angles, an equilateral triangle is also acute
5. Acute Triangle
3 Acute Angles
6. Parts of a Triangle
Each of the three points joining the sides of a triangle is a vertex.(plural:vertices).A, B and C are vertices.
Two sides sharing a common vertex are adjacent sides.
The third is the side opposite an angle
adjacent
opposite
adjacent
7.

Red represents the hypotenuse of a right triangle.The sides that form the right angle are the legs.

Right Triangles
leg
Hypotenuse
leg
8. Finding Angle Measures
Corollary to the triangle sum theorem
The acute angles of a right triangle are complementary.
m A + m B =90
2X
X
9. Finding Angle Measures
B
2X
X + 2x = 90
3x = 90
X = 30
So m A = 30 and the m B=60
X
A
C
10. Angle Bisector

A ray that divides an angle into 2 congruent adjacent angles.

BD is an angle bisector of A
B
D
C
11. Reason:
Given
Def. Cong. Angles
Def. Cong. Angles
Transitive property
Def. Cong. Angles
Statement:
A B, B C
mA= mB
mB= mC
mA= mC
A C
Ex. 1:Transitive Property of Angle Congruence
12. Using the Transitive Property
Given: m3 40, 1 2, 2 3
Prove: m1 =40
13. All right angles are congruent.
Example 3:Proving Theorem 2.3
Given: 1 and 2 are right angles
Prove: 1 2
Theorem 2.0
14. Theorem 2.1:Congruent Supplements.If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
If m1+m2 = 180 AND m2+m3 = 180, then1 3.
Properties of Special Pairs of Angles
15. Theorem 2.2:If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent.
If m4+m5 = 90 AND m5+m6 = 90, then4 6.
Congruent Complements Theorem
16. Median of a Triangle
A median of a triangle is a segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side.For instance in ABC, shown at the right, D is the midpoint of side BC.So, AD is a median of the triangle
17. The three medians of a triangle are concurrent (they meet).The point of concurrency is called the center of the triangle.The center, labeled P in the diagrams in the next few slides are ALWAYS inside the triangle.
Center of the Triangle
18. Comparing Measurements of a Triangle

After practicing a few exercises, you may have discovered a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles.