■ An acute angle is an angle that measures between 0 and 90 degrees. ■ A right angle is an angle that measures exactly 90°. A right angle is symbolized by a square at the vertex. ■ An obtuse angle is an angle that measures more than 90°, but less than 180°. ■ A straight angle is an angle that measures 180°. Thus, both of its sides form a line. Straight Angle 180° Obtuse Angle Right Angle Symbol A cute Angle – THEA MATH REVIEW– 150 SPECIAL ANGLE PAIRS ■ Adjacent angles are two angles that share a common vertex and a common side. There is no numerical relationship between the measures of the angles. ■ A linear pair is a pair of adjacent angles whose measures add to 180°. ■ Supplementary angles are any two angles whose sum is 180°. A linear pair is a special case of supplemen- tary angles. A linear pair is always supplementary, but supplementary angles do not have to form a linear pair. ■ Complementary angles are two angles whose sum measures 90 degrees. Complementary angles may or may not be adjacent. Example Two complementary angles have measures 2x° and 3x + 20°. What are the measures of the angles? Since the angles are complementary, their sum is 90°. We can set up an equation to let us solve for x: 2x + 3x + 20 = 90 5x + 20 = 90 5x = 70 x = 14 Substituting x = 14 into the measures of the two angles, we get 2(14) = 28° and 3(14) + 20 = 62°. We can check our answers by observing that 28 + 62 = 90, so the angles are indeed complementary. 50 ˚ 40 ˚ 50 ˚ Adjacent complementary angles Non-adjacent complementary angles 40 ˚ 70 ˚ 110 ˚ 70 ˚ 110 ˚ Linear pair (also supplementary) Supplementary angles (but not a linear pair) 1 2 1 2 Adjacent angles ∠1 and ∠2 Non-adjacent angles ∠1 and ∠2 – THEA MATH REVIEW– 151 Example One angle is 40 more than 6 times its supplement. What are the measures of the angles? Let x = one angle. Let 6x + 40 = its supplement. Since the angles are supplementary, their sum is 180°. We can set up an equation to let us solve for x: x + 6x + 40 = 180 7x + 40 = 180 7x = 140 x = 20 Substituting x = 20 into the measures of the two angles, we see that one of the angles is 20° and its supplement is 6(20) + 40 = 160°. We can check our answers by observing that 20 + 160 = 180, prov- ing that the angles are supplementary. Note: A good way to remember the difference between supplementary and complementary angles is that the letter c comes before s in the alphabet; likewise “90” comes before “180” numerically. ANGLES OF INTERSECTING LINES Important mathematical relationships between angles are formed when lines intersect. When two lines intersect, four smaller angles are formed. Any two adjacent angles formed when two lines intersect form a linear pair, therefore they are supplemen- tary. In this diagram, ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, and ∠4 and ∠1 are all examples of linear pairs. Also, the angles that are opposite each other are called vertical angles. Vertical angles are angles who share a vertex and whose sides are two pairs of opposite rays. Vertical angles are congruent. In this diagram, ∠1 and ∠3 are vertical angles, so ∠1 ≅∠3; ∠2 and ∠4 are congruent vertical angles as well. Note: Vertical angles is a name given to a special angle pair. Try not to confuse this with right angle or per- pendicular angles, which often have vertical components. 2 1 3 4 – THEA MATH REVIEW– 152 Example Determine the value of y in the diagram below: The angles marked 3y + 5 and 5y are vertical angles, so they are congruent and their measures are equal. We can set up and solve the following equation for y: 3y + 5 = 5y 5 = 2y 2.5 = y Replacing y with the value 2.5 gives us the 3(2.5) + 5 = 12.5 and 5(2.5) = 12.5. This proves that the two vertical angles are congruent, with each measuring 12.5°. PARALLEL LINES AND TRANSVERSALS Important mathematical relationships are formed when two parallel lines are intersected by a third, non-parallel line called a transversal. In the diagram above, parallel lines l and m are intersected by transversal n. Supplementary angle pairs and vertical angle pairs are formed in this diagram, too. Supplementary Angle Pairs Vertical Angle Pairs ∠1 and ∠2 ∠2 and ∠4 ∠1 and ∠4 ∠4 and ∠3 ∠3 and ∠1 ∠2 and ∠3 ∠5 and ∠6 ∠6 and ∠8 ∠5 and ∠8 ∠8 and ∠7 ∠7 and ∠5 ∠6 and ∠7 2 1 3 4 6 5 7 8 l m n 5y 3y + 5 – THEA MATH REVIEW– 153 Other congruent angle pairs are formed: ■ Alternate interior angles are angles on the interior of the parallel lines, on alternate sides of the transver- sal: ∠3 and ∠6; ∠4 and ∠5. ■ Corresponding angles are angles on corresponding sides of the parallel lines, on corresponding sides of the transversal: ∠1 and ∠5; ∠2 and ∠6; ∠3 and ∠7; ∠4 and ∠8. Example In the diagram below, line l is parallel to line m. Determine the value of x. The two angles labeled are corresponding angle pairs, because they are located on top of the parallel lines and on the same side of the transversal (same relative location). This means that they are con- gruent, and we can determine the value of x by solving the equation: 4x + 10 = 8x – 25 10 = 4x – 25 35 = 4x 8.75 = x We can check our answer by replacing the value 8.75 in for x in the expressions 4x + 10 and 8x – 25: 4(8.75) + 10 = 8(8.75) – 25 45 = 45 Note: If the diagram showed the two angles were a vertical angle pair or alternate interior angle pair, the prob- lem would be solved in the same way. 4x + 10 l m 8x – 25 n – THEA MATH REVIEW– 154 Area, Circumference, and Volume Formulas Here are the basic formulas for finding area, circumference, and volume. They will be discussed in detail in the following sections. Triangles The sum of the measures of the three angles in a triangle always equals 180 degrees. a b c a + b + c = 180° Circle Rectangle Triangle r l w h b A = lw A = 1 _ 2 bh C = 2πr A = πr 2 Cylinder Rectangular Solid h l V = πr 2 h w r h V = lwh C = Circumference A = Area r = Radius l = Length w = Width h = Height v = Volume b = Base – THEA MATH REVIEW– 155 . ∠2 and ∠4 ∠1 and ∠4 ∠4 and 3 3 and ∠1 ∠2 and 3 ∠5 and ∠6 ∠6 and ∠8 ∠5 and ∠8 ∠8 and ∠7 ∠7 and ∠5 ∠6 and ∠7 2 1 3 4 6 5 7 8 l m n 5y 3y + 5 – THEA MATH REVIEW– 1 53 Other congruent angle pairs. intersect form a linear pair, therefore they are supplemen- tary. In this diagram, ∠1 and ∠2, ∠2 and 3, 3 and ∠4, and ∠4 and ∠1 are all examples of linear pairs. Also, the angles that are opposite each. pairs of opposite rays. Vertical angles are congruent. In this diagram, ∠1 and 3 are vertical angles, so ∠1 ≅ 3; ∠2 and ∠4 are congruent vertical angles as well. Note: Vertical angles is a name