Lines Lesson 1
Transversals and Angles Interior angles: Exterior angles: Alternate Interior angles: Alternate Exterior angles: Corresponding angles:
Example 1 Classify each pair of angles in the figure as alternate interior, alternate exterior, or corresponding. corresponding angles alternate exterior angles
Got it? 1 Classify the relationship between alternate interior angles
Missing Angle Measures m2 = 130˚ because and are supplementary. If = 50˚, find , , and m = 50˚ because and 3 are vertical angles. m4 = 130˚ because and are supplementary.
Example 2 A furniture designer built the bookcase shown. Line a is parallel to line b. If m2 = 105˚, find m6 and m. Justify your answer. Since 2 and 6 are supplementary, the m6 = 75˚. Since 6 and 3 are interior angles, so the m3 is 75˚.
Got it? 2 Find the measure of angle 4. 105º; 2 and 4 are corresponding angles, so their measures are equal.
Example 3 In the figure, line m is parallel to line n, and line q is perpendicular to line p. The measure of 1 is 40˚. What is the measure of 7. Since 1 and 6 are alternate exterior angles, m6 = 40˚. Since 6, 7, and 8 form a straight line, the sum is 180˚. 40 + 90 + m7 = 180 So m7 is 50˚.
Geometric Proof Lesson 2
Deductive vs. Inductive Reasoning Every time Bill watches his favorite team on TV, the team loses. So, he decides to not watch the team play on TV. Deductive Reasoning In order to play sports, you need to have a B average. Simon has a B average, so he concludes that he can play sports. All triangles have 3 sides and 3 angles. Mariah has a figure with 3 sides and 3 angles so it must be a triangle. Inductive Reasoning After performing a science experiment, LaDell concluded that only 80% of tomato seeds would grow into plants.
The Proof Process STEP 1: List the given information, or what you know. Draw a diagram if needed. STEP 2: State what is to be proven. STEP 3: Create a deductive argument by forming a logical chain of statements linking the given information. STEP 4: Justify each statement with definitions, properties, and theorems STEP 5: State what it is you have proven.
Vocabulary A proof is a logical argument where each statement is justified by a reason. A paragraph proof or informal proof involves writing a paragraph. A two-column proof or formal proof contains statements and reason organized in two columns. Once a statement has been proven, it is a theorem.
Example 1 – Paragraph Proof The diamondback rattlesnake has a diamond pattern on its back. An enlargement of the skin is shown. If m1 = m4, write a paragraph proof to show that m = m3. Given: m1 = m4 Prove: m = m3 Proof: m1 = m2 because they are vertical angles. Since m1 = m4, and m = m4. The measure of angle 3 and 4 are the same since they are vertical angles. Therefore, m = m3.
Got it? 1 Refer to the diagram shown. AR = CR and DR = BR. Write a paragraph proof to show that AR + DR = CR + BR. Given: AR = ___________ and DR = ____________. Prove: _________________ = CR + BR. Proof: You know that AR = CR and DR = BR. AR + DR = CR + BR by the _____________ Property of Equality. So, AR + DR = CR + BR by ___________________. CR BR AR + DR Addition substitution
Example 2 Write a two-column proof to show that if two angles are vertical angles, then they have the same measure. Given: lines m and n intersect; 1 and 3 are vertical. Prove: m1 = m3 Reasons Statements • Lines m and n intersect; 1 and 3 are vertical. • 1 and 2 are a linear pair and 3 and 2are a linear pair. • m1 and m2 = 180˚ • m3 and m2 = 180˚ • d. m1 and m2 = m3 and m2 • e. m1 = m3 Given Definition of linear pair Definition of supplemental angles Substitution Subtraction Property of Equality
Got it? 2 The statements for a two-column proof to show that if mY = mZ, then x =100 are given below. Complete the proof by providing the reasons. Reasons Statements • mY = mZ, • mY = 2x – 90 • mZ = x + 10 • b. 2x – 90 = x + 10 • c. x – 90 = 10 • d. x = 100 Given Substitution Subtraction Property of Equality Addition Property of Equality
Angles of Triangles Lesson 3
Real-World Link 1. What is true about the measures of 1 and 2? Explain. They are equal because they are alternate interior angles. 2. What is true about the measures of 3 and 4? They are equal because they are alternate exterior angles. 3. What kind of angle is formed by 1, 5, and 3? Write an equation representing the relationship between the 3 angles. Straight angle = 1 + 5, + 3 = 180˚ 4. Draw a conclusion about ΔABC. The sum of the angles in ΔABC is 180˚.
Angle Sum of a Triangle Words: The sum of the measures of the interior angles of a triangle is 180˚. Symbols: x + y + z = 180˚. Model:
Example 1 Find the value of x in the Antigua and Barbuda flag. x + 55 + 90 = 180 x + 145 = 180 x = 35 The value of x is 35.
Got it? 1 In ΔXYZ, if mX = 72˚ and mY = 74˚, what is mZ? 72 + 74 + Z = 180 146 + Z = 180 Z = 34 The measure of angle Z is 34 degrees.
Example 2 The measures of the angles of ΔABC are in the ratio 1:4:5. What are the measures of the angles? Let x represent angle A, 4x angle B, and 5x angle C x + 4x + 5x = 180 10x = 180 x = 18 Angle A = 18˚ Angle B = 18(4) = 72˚ Angle C = 18(5) = 90˚
Got it? 2 The measures of the angles of ΔLMN are in the ratio 2:4:6. What are the measures of the angles? Let x represent angle L, 4x angle M, and 5x angle N 2x + 4x + 6x = 180 12x = 180 x = 15 Angle L = 15(2) = 30˚ Angle M = 15(4) = 60˚ Angle N = 15(6) = 90˚
Exterior Angles of a Triangle Words: The measure of an exterior angle is equal to the sum of the measures of its two remote interior angles. Symbols: mA + mB = m1 Model:
Interior and Exterior Angles Each exterior angle of the triangle has two remote interior angles that are not adjacent to the exterior angle. 4 1 6 2 4 is an exterior angle. It’s two remote angles are 2 and 3. m4 = m2 + m3 interior exterior 3 5
Example 3 First Way: Angle 4 is the exterior angle with angle 2 and angle K as the remote interior. 2 + K = 4 2 + 90 = 135 2 = 45˚ Suppose m4 = 135˚. Find the measure of 2. Second Way: 4 and 1 are supplementary, so they equal 180˚. 4 + 1 = 180 135 + 1 = 180 1 = 45 1 + 2 + K = 180 45 + 2 + 90 = 180 2 = 45˚
Got it? 3 Suppose m5 = 147˚. Find m1. m1 = 57˚
Polygons and Angles Lesson 4
Real-World Link A polygon is a closed figure with three of more line segments. List the states that are in a shape of a polygon. North Dakota New Mexico Utah Wyoming Colorado
Interior Angle Sum of a Polygon Words: The sum of the measures of the interior angles of a polygon is (n – 2)180, where n is the number of sides. Symbols: S = (n – 2)180 Regular Polygons – an equilateral (all sides are the same) and a equiangular (all angles are the same)
Example 1 Find the sum of the measures of the interior angles of a decagon. S = (n -2)180 S = (10 – 2)180 S = (8)180 S = 1,440 The sum of the interior angles of a 10-sided polygon is 1,440˚.
Got it? 1 Find the sum of the measures of the interior angles of each polygon. a. hexagon 720˚ b. octagon 1,080˚ c. 15-gon 2, 340˚
Example 2 Each chamber of a bee honeycomb is a regular hexagon. Find the measure of an interior angle of a regular hexagon. STEP 1: Find the sum of the measures of angle. S = (n – 2)180 S = (6 – 2)180 S = (4)180 S = 720˚ STEP 2: Divide 720 by 6, since there are six angles in a hexagon. 720˚÷ 6 = 120 Each angle in a hexagon is 120˚
Got it? 2 Find the measure of one interior angle in each regular polygon. Round to the nearest tenth if necessary. a. octagon 135˚ b. heptagon 128.6˚ c. 20-gon 162˚
Exterior Angles of a Polygon Words: The sum of the measures of the exterior angles, one at each vertex, is 360˚. Symbols: m1 + m2 + m3 + m4 + m5 = 360˚ Model: Examples:
Example 3 Find the measure of an exterior angle in a regular hexagon. A hexagon has a 6 exterior angles. 6x = 360 x = 60 Each exterior angle is 60˚.
Got it? 3 Find the measure of an exterior angle in a regular polygon. a. triangle 120 ˚ b. quadrilateral 90 ˚ c. octagon 45 ˚
The Pythagorean Theorem Lesson 5
Pythagorean Theorem Words: In a right triangle, the sum of the squares of the legs equal the square of the hypotenuse. Symbols: a2 + b2 = c2 Model: c a b
Example 1 Find the missing length. Round to the nearest tenth. a2 + b2 = c2 92 + 122 = c2 81 + 144 = c2 225 = c2 = c c = 15 and -15 The equation has two solutions, -15 and 15. However, the length of the side must be positive. The hypotenuse is 15 inches long. c 12 in 9 in
Example 2 Find the missing length. Round to the nearest tenth. a2 + b2 = c2 82 + b2 = 242 64 + b2 = 576 64 – 64 + b2 = 576 - 64 b2 = 512 b = b 22.6 or -22.6 The length of leg b is 22.6 cm long. 24 cm b 8 cm
Got it? 1 and 2 Find the missing length. Round to the nearest tenth if necessary. a. b. The length of the hypotenuse is 30 yards long. The length of leg a is 10.5 cm long.
Converse of Pythagorean Theorem STATEMENT: If a triangle is a right triangle, then a2 + b2 = c2. CONVERSE: If a2 + b2= c2, then a triangle is a right triangle. The converse of the Pythagorean Theorem is also true.
Example 3 The measures of three sides of a triangle are 5 inches,12 inches and 13 inches. Determine whether the triangle is a right triangle. a2 + b2 = c2 52 + 122 = 132 25 + 144 = 169 169 = 169 The triangle is a right triangle.
Got it? 3 Determine if these side lengths makes a right triangle. a. 36 in, 48 in, 60 in b. 4 ft, 7ft, 5ft no yes
Use the Pythagorean Theorem Lesson 6
Example 1 Write an equation that can be used to find the length of the ladder. Then solve. Round to the nearest tenth. a2 + b2 = c2 8.752 + 182 = x2 76.5625 + 324 = x2 400.5625 = x2 = x 20.0 x The ladder is about 20 feet.