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Exploring Isosceles Triangles

Learn about the properties of isosceles triangles, the sum of angle measures in a triangle, and how to prove the Triangle Sum Conjecture. Practice solving problems and identifying isosceles triangles.

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Exploring Isosceles Triangles

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  1. Do Now • Take out your compass and a protractor. • Look at the new seating chart and find your new seat. • Classify this triangle: • By angles • By side lengths • On a piece of paper draw a triangle. (It can be acute, right, or obtuse.) Make it big enough to measure the angles.

  2. Isosceles Triangles

  3. TODAY’S OBJECTIVES • Discover the relationship between the base angles of an isosceles triangle. • Explain the sum of the measures of the angles of a triangle. • Write a paragraph proof. • Use problem solving skills.

  4. YOU MAY ASSUME THAT… • Lines are straight • If two lines intersect, they intersect at a point.

  5. DO NOT ASSUME THAT… • Lines are parallel unless they are marked parallel, even if they “look” parallel • Lines are perpendicular unless they are marked perpendicular, even if they “look” perpendicular • Pairs of angles, segments, or polygons are congruent unless they are marked congruent, even if they “look” congruent.

  6. The triangle sum: Investigation • On a piece of paper, draw a triangle. (Make sure your group has at least one obtuse and one acute triangle.) • Measure all three angles as accurately as possible. • Find the sum of the measures of the three angles. Compare with your group. • Mark your angles A, B, and C. Cut out the triangle. • Tear off the three angles. Arrange them so their vertices meet at a point. How does this arrangement show the sum of the angle measures?

  7. Triangle Sum Conjecture • The sum of the measures of the angles in every triangle is___. • 180o . • Based on what type of reasoning? • Inductive. • Can we prove it using deductive reasoning? • Let’s prove it!

  8. Proof of Triangle Sum Conjecture • As a group, explain why the Triangle Sum Conjecture is true by writing a paragraph proof (a deductive argument that uses written sentences to support its claims with reasons). • Hints to get started: • What are you trying to prove? • How are the angles related? • Mark your diagram. • How can you use the information you have to prove that the Triangle Sum Conjecture is true for every triangle? • Remember what you can and cannot assume.

  9. Practice • Solve for p and q.

  10. Properties of Isosceles Triangles • Two sides are congruent

  11. Base Angles in an Isosceles Triangle: Investigation • Draw an angle. Label it C. This will be the vertex angle of your isosceles triangle. • Place a point A on one ray. Using your compass, copy segment CA onto the other ray and mark point B so that CA=CB. • Draw AB. • How do you know ΔABC is isosceles? • Name the base and the base angles. • Use your protractor to measure the base angles. What do you notice?

  12. Isosceles Triangle Conjecture • If a triangle is isosceles, then ____________________________. • it’s base angles are congruent. • Is the converse true? • Let’s find out.

  13. Converse: Investigation • Draw a segment and label it AB. Draw an acute angle at A. • Copy A at point B on the same side of the segment. • Label the intersection of the two rays point C. • Use your compass to compare the lengths of AC and BC. What do you notice?

  14. Converse of the Isosceles Triangle Conjecture • If a triangle has two congruent angles, then _______________. • it is an isosceles triangle.

  15. Practice • Find the measure of T.

  16. Stations • Collaborative: Start your group project. • Independent: Get familiar with McGraw Hill • Direct: Practice.

  17. What’s wrong with this picture? • A

  18. Practice • Solve for r, s and t.

  19. Practice • The perimeter of ΔQRS is 344. • mQ= • QR=

  20. TODAY’S OBJECTIVES • Discover the relationship between the base angles of an isosceles triangle. • Explain the sum of the measures of the angles of a triangle. • Write a paragraph proof. • Use problem solving skills.

  21. Exit Slip For each question, show your work and explain your reasoning. • Find x (above). • mA= • a= • The perimeter of ΔABC=

  22. Honors Exit Slip • Find x (above). Explain your reasoning. • mA= • The perimeter of ΔABC= • Use the diagram below to explain why ΔPQR is isosceles.

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