4.1 triangles and angles. definition: triangle a triangle is a figure formed by three segments...

4.1 Triangles and Angles

Definition: Triangle

A triangle is a figure formed by three segments joining three noncollinear points.

Triangles are always classified in two ways:

1. By Sides

2. By Angles

Subtending Angles & Sides• When two sides are equal then their subtending angles

must be equal. • Likewise, when two angles are equal then their

subtending sides must be equal.

Angle A subtends Side BCAngle B subtends Side ACAngle C subtends Side AB

If AB = BC, then which two angles must be equal?

If <B = <C, then which two sides must be equal?

A

B

C

Classification by SidesEquilateral Triangle = 3 congruent sides

Isosceles Triangle = At least 2 congruent sides

Scalene Triangle = No congruent sides

Classification by AnglesAcute Triangle: Three Acute Angles (all measure less than 90o)

Equiangular Triangle: Three congruent angles (each measures 60o)

Right Triangle; One Right Angle (90o)

Obtuse Triangle: One obtuse Angle (more than 90o) and two acute angles (less than 90o)

Example 1: Classifying Triangles

When you classify a triangle, you must give TWO classifications (sides & angles.)

How would you classify triangle ABC?

Sides: _____________

Angles: ____________

Example 2: Classifying Triangles

How would you classify this triangle?

Sides: _____________

Angles: ____________

Example 3: Classifying Triangles

How would you classify this triangle?

Sides:

Angles:

Special Labels for Right and Isosceles Triangles

Hypotenuse

Leg

Leg

Base

Leg Leg

By extending the sides we create Interior and Exterior Angles

Interior Angles are “inside” the triangle. Exterior Angles are “outside the triangle. Label each interior angle with an “I” and each exterior angle with an “E.” When an interior angle and an exterior angle are together they make a straight line and sum to 180o.

Triangle Sum Theorem

• The three angles inside of a triangle must always add to 180o.

180m A m B m C

If m<A = 75o and m<B = 85o, what must the measure of angle C be?

A special and useful property!

• The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent interior angles

1A

B

C

1m m A m B If m<A = 90o and m<B = 45o, then what is m<1?

If m<1 = 102o and m<B = 55o, then what is m<A?