Angles and their measurements

Degrees: Measuring Angles

We measure the size of an angle using degrees. Example: Here are some examples of angles and their degree measurements.

Acute Angles

An acute angle is an angle measuring between 0 and 90 degrees.

Example:

                                                             

  

Obtuse Angles

An obtuse angle is an angle measuring between 90 and 180 degrees.

Example:

                                                              

   

Right AnglesA right angle is an angle measuring 90 degrees.

Examples:

                                      

Complementary AnglesTwo angles are called complementary angles if the sum of their degree measurements equals 90 degrees.

Example:

These two angles are complementary.

                                                      

                        

These two angles can be "pasted" together to form a right angle!

                        

Supplementary AnglesTwo angles are called supplementary angles if the sumof their degree measurements equals 180 degrees.

Example:

These two angles are supplementary.

                                                                      

These two angles can be "pasted" together to form a straight line!

                                                

Vertical Angles

For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Angle BEC and angle AED are also vertical angles. Vertical angles have the same degree measurement.                                    

Review

State whether the following are acute, right, or obtuse.1.

2.

3.

4.

5.

??

acute obtuseright

obtuseacute

Complementary and Supplementary

1. Two angles are complementary. One measures 65 degrees.

2. Two angles are supplementary. One measures 140 degrees.

Find the missing angle.

Answer : 25

Answer : 40

Complementary and Supplementary

Find the missing angle. You do not have a protractor.Use the clues in the pictures.

1.2.x

55 165x

X=35 X=15

Vertical Angles

Find the missing angle. You do not have a protractor.

Use the clues in the pictures.

58x

X=58

Triangles

Triangles

• A triangle is a 3-sided polygon. Every triangle has three sides and three angles.

• When added together, the three angles equal 180°.

Acute Triangles

• A triangle in which all three angles are less than 90°.

Obtuse Triangles

• An obtuse triangle is a triangle in which one of the angles is greater than 90°.

Right Triangles

• A right triangle is triangle with an angle of 90°.

Scalene Triangles

• A triangle with three unequal sides.

Isosceles Triangles

• An isosceles triangle is a triangle with (at least) two equal sides.

Equilateral Triangles

• An equilateral triangle is a triangle with all three sides of equal length.

Equiangular Triangles

• An equiangular triangle is a triangle which has three equal angles.

Congruent Triangles

• Congruent triangles are triangles whose corresponding angles and sides are congruent.

• They are exactly the same size and shape.

The Hypotenuse

• The hypotenuse of a right triangle is the triangle's longest side, i.e., the side opposite the right angle.

The Pythagorean Theorem

• For a right triangle with legs a and b and hypotenuse c, a2+b2=c2.

Quick Quiz (all together!)

Question 1

• If a triangle has three unequal sides it is a ______ triangle.– A) Scalene– B) Obtuse– C) Isosceles

• Answer:– A) Scalene

Question 2

• A triangle which has three equal angles is a ______ triangle.– A) Acute– B) Equilateral– C) Equiangular

• Answer:– C) Equiangular

Question 3

• A triangle with (at least) two equal sides is a ______ triangle.– A) Scalene– B) Isosceles– C) Congruent

• Answer:– B) Isosceles

Question 4

• Two triangles that are equal are called ______ triangles– A) Acute – B) Obtuse – C) Congruent

• Answer:– C) Congruent

Question 5

• A triangle with all three sides of equal length is a ______ triangle.– A) Equilateral– B) Right– C) Equiangular

• Answer:– A) Equilateral

Question 6

• A triangle with an angle of 90° is a ______ triangle.– A) Acute– B) Right– C) Congruent

• Answer:– B) Right

Question 7

• A triangle in which all three angles are less than 90° is a ______ triangle.– A) Right– B) Acute– C) Scalene

• Answer:– B) Acute

Question 8

• A triangle in which one of the angles is greater than 90° is a ______ triangle.– A) Obtuse– B) Isosceles– C) Acute

• Answer:– A) Obtuse

Question 9

• What is the Hypotenuse of a right triangle?– A) The smallest angle.– B) The side opposite the

right angle.– C) The right angle.

• Answer:– B) The side opposite the

right angle.

Question 10

• The Pythagorean Theorem states …– A) r2

– B) (a+b)2/2=c– C) a2+b2=c2

• Answer:– C) a2+b2=c2

Question 11

• Use the Pythagorean Theorem to solve the following:– a = 5– b = 12– c = ?

• Answer:– c = 13