Geometry Notes Chapter 4: Triangles

Transcription

1 Geometry Notes Chapter 4: Triangles Name Date Assignment Questions I have Day 1 Section 4.1: Triangle Sum, Exterior Angles, and Classifying Triangles Day 2 Assign: Finish Ch. 3 Review Sheet, WS 4.1 Section 4.2: Triangle Inequalities and Base Angle Theorem Day 3 Assign: WS 4.2 Sections : Review Day Section 4.3: Converse of Pythagorean Quiz Day 4 Assign: Review WS Section 4.4: Midsegments and Special Triangle Segments Day 5 Assign: WS 4.4 Chapter 4 Review Day Day 6 Chapter 4 Test Day

2 4.1 Triangle Sum Properties Triangle Sum = Classifying Triangles Scalene: by Sides Acute: by Angles Isosceles: Right: Equilateral: Obtuse: Equiangular: Find the value of x and determine the kind of

3 3. Interior & Exterior Angles of a Triangle Interior Angles of a Triangle: Exterior Angles of a Triangle: forms a linear pair with an interior angle of the triangle Remote Interior Angles of a Triangle: given an exterior angle, these are the inside far away angles exterior angle remote interior angles

4 Given: ABC Prove: 1. ABC m 2 + m 3 + m 4 = m 4 + m 1 = m 2 + m 3 + m 4 = m 4 + m m 2 + m 3 = m 1 5. Exterior Angle Theorem Find the value of the indicated variables

5 3. 4. AB // CD 5. MPN is a right angle 6. 7.

6 4.2 Inequalities in a Triangle Given an obtuse scalene triangle Which side is the longest? 2. Which angle is the biggest? 3. Which side is the shortest? 3. Which angle is the smallest? Order the sides from shortest to longest. Order the angles from smallest to biggest.

7 Given three sets of numbers to represent side lengths, which of these will make a triangle? a. 2, 2, 5 b. 3, 5, 2 c. 5, 4, 2 Identifying Possible Triangles Is it possible to construct a triangle with the given side lengths? 1. 6, 7, , 6, , 34, , 120, 125 Find Possible Side Lengths A triangle has one side of length 12 and another of length 8. Describe the possible lengths of the third side. Describe the possible lengths of the third side of the triangle given the lengths of the other two sides inches, 12 inches 2. 3 meters, 4 meters feet, 7 yards 4. 2 feet, 40 inches

8 Isosceles and Equilateral Triangles: If two sides of a triangle are congruent, then the angles opposite them are congruent. If two angles of a triangle are congruent, then the sides opposite them are congruent. If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral. Determine the values of x, y and z

9 4.3 Pythagorean Theorem Converse (Hypotenuse) 2 = (Leg 1) 2 + (Leg 2) 2 Pythagorean Theorem Solve for the value of x. Answers must be in simplified radical form. 1. x = 2. x = 3. x = Converse to the Pythagorean Theorem Given: a, b, and c are lengths of the sides of a triangle, where c is the longest side. If c 2 = a 2 + b 2, then the triangle is a triangle. If c 2 < a 2 + b 2, then the triangle is a triangle. If c 2 > a 2 + b 2, then the triangle is a triangle. What kind of triangle??

10 Putting it all together. First, decide whether the set of numbers can represent the side lengths of a triangle. Then, if they can, classify the triangle as Right, Acute, or Obtuse , 38, , 11, , 20.5, , 37.5, Midsegment Theorem and Coordinate Proof Warm Up For the following points: A (0, 10), B (12, 0), and C (0, 0) 1. Find AB. 2. Find the midpoint of CA. 3. Find the midpoint of. AB 4. Find the slope of AB. Midsegment: Every triangle has midsegments. Use a ruler to draw the three midsegments on the following triangle.

11 Explore: 1) Use coordinate geometry to find the midpoints of sides AB and AC. Name the midpoint of AB point D. Name the midpoint of AC point E. 2) Draw the midsegment. DE 3) Find the slopes of DE and BC. 4) Find the lengths of DE and BC. 5) Make a conjecture about the relationships between the midsegments and the opposing sides. Midsegment Theorem The midsegments are to the third side and as long as that side. GUIDED PRACTICE DE is a midsegment of ABC. Find the value of x.

12 In JKL, JR RK, KS SL, andjt. TL 4. RS 5. ST 6. KL 7. SL 8. JR 9. JT Use GHJ, where D, E, and F are midpoints of the sides. 10. If DE = 4x + 5 and GJ = 3x + 25, what is DE? 11. If EF = 2x + 7 and GH = 5x 1, what is EF? 12. If HJ = 8x 2 and DF = 2x + 11, what is HJ? A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a. 1. Use a straight-edge and protractor to construct the perpendicular bisector of. AB 2. Plot a point C on the perpendicular bisector. 3. Use a ruler to measure AC and BC. 4. Write a conjecture about a point on the perpendicular bisector.

13 Perpendicular Bisector Theorem A perpendicular bisector of a triangle is a segment to a side at its The is equidistant from the. *** Any point on the perpendicular bistector is to the endpoints. 1. BD is the perpendicular bisector of AC. 2. Is KH Find AD. the perpendicular bisector of RT? If so, find RT. An is a ray that divides an angle into two congruent adjacent angles. Use a protractor and straightedge to draw the angle bisector.

14 Angle Bisector Theorem A point is on the bisector of an angle if and only if it is from the two sides of the angles. e. Solve for x. 1. x = 2. x = Median & Altitude The of a triangle is a segment from a vertex to the midpoint of the opposite side. The of a triangle is a segment from the vertex of a triangle drawn perpendicular to the opposite side.

15 Special Segments in a Triangle: Perpendicular Bisector Angle Bisector Median Altitude