Section 1.5 Triangles Notes Goal of the lesson: Explore the properties of triangles using Geometer s Sketchpad Define and classify triangles and their related parts Practice writing more definitions Learn more vocabulary related to triangles Warm up: 1. Use the coordinate plane below to locate point R so that is an isosceles right triangle. Write at least three possible coordinates for point R. 2. Sketch and label two different isosceles triangles with perimeter 4a + b. 3. Sketch an isosceles acute triangle with base AC and vertex angle B. 4. Use the diagram at the right and proper notation to name a pair of: a. Parallel lines b. Perpendicular line segments c. Congruent line segments d. Vertical angles e. Linear pair angles
Assumptions in geometry: Things you MAY assume: Lines, line segment, rays are straight If two lines intersect, they intersect at one point That points on a line are collinear All points shown in a diagram are coplanar unless planes are drawn to show that they are noncoplanar Things you MAY NOT assume: In the diagrams below, which pairs of lines are perpendicular? Which pairs of lines are parallel? Which pairs of triangles is congruent?
Geometer s Sketchpad Investigation: Triangles In this activity you ll experiment with both ordinary triangles and triangles that were constructed with constraints. The constraints limit what you can change in the triangle when you drag so that certain relationships among angles and sides always hold. By observing these relationships, you will classify the triangles. Sketch Step 1: Open the sketch Triangles.gsp. You can find the sketch by clicking on the sub page Chapter 1 Sketch pad activities. Step 2: Drag different vertices of each of the four triangles to observe and compare how the triangles are constrained. Investigate: 1. Which of the triangles seems the least constrained (the easiest to change by dragging)? Explain and be sure to use proper to notation when referring to the triangle. 2. Which of the triangles seems the most constrained (the hardest to change by dragging)? Explain and be sure to use proper to notation when referring to the triangle. 3. Recall the definitions of acute, obtuse, and right angles. These terms are also used to classify triangles. Measure the three angles in. (To measure an angle, select only three points on the angle, with the vertex as the second selection, choose Measure Angle OR highlight the two sides of the angle and the vertex and choose Measure Angle) Determine whether each angle is acute or obtuse. Now drag a vertex of your triangle. How many acute angles can a triangle have?
4. How many obtuse angles can a triangle have? 5. can be either an obtuse triangle or an acute triangle. One other triangle in the sketch can also be either acute or obtuse. Which triangle is it? Name it properly. 6. Which triangle is always a right triangle, no matter what you drag? Be sure to measure the angles. Name it properly. 7. Which triangle is always an equiangular triangle, no matter what you drag? Be sure to measure the angles. Name it properly. 8. Scalene, isosceles, and equilateral are terms used to classify triangles by relationships among their sides. Measure the lengths of the three sides of. To measure the lengths, click on the segments of the triangle and Measure Lengths. If none of the side lengths are equal, the triangle is a scalene triangle. If two or more of the sides are equal in length, the triangle is isosceles. If all three sides are equal in length, it is equilateral. (But since isosceles is two or MORE equal sides, a triangle with three sides equal is also isosceles). Because has no constraints, it can be any type of triangle. But which type of triangle is most of the time? 9. Name a triangle other than that is scalene most of the time. Name it properly. 10. Which triangle (or triangles) is (are) always isosceles, no matter what you drag? Name it (them) properly. 11. Which triangle is always equilateral, no matter what you drag?
In Questions 12-16, tell whether the triangle described is possible or not possible. To check your answer, try manipulating the triangles in the sketch to make one that fits the description. If the triangle is possible, sketch an example on your paper and mark the appropriate sides or angles. 12. Obtuse isosceles triangle 13. Acute right triangle 14. Obtuse equiangular triangle 15. Isosceles right triangle 16. Acute scalene triangle In the Geometer s Sketch pad investigation, you explored all types of triangles. With your group members, write the definitions of the following triangles. Right triangle: Acute triangle: Obtuse triangle: Scalene triangle: Equilateral triangle: Isosceles triangle:
In an isosceles triangle, the vertex angle is the angle between the two sides of equal length. The side opposite the vertex angle is the base of the isosceles triangle and the two angles opposite the two congruent sides are called the base angles. Try these problems and discuss with your group mates to make sure you are all correct. 1. Sketch and label the figure with marks that is an isosceles acute triangle ACT with AC = CT. 2. Return to the warm up problems and complete them if you have not already. 3. Use the coordinate plane below to locate point L so that is an isosceles triangle. Write at least three possible coordinates for point L. Homework: Section 1.5 HW in the textbook: Review p. 59 61 and do p. 62 63 #1-4, 6, 7, 13, 18-20 and finish the notes (if you did not finish them in class)