Lesson 10 Centers of a Triangle Practice Understanding

Ready

Figure 1 has been rotated $180 °$ about the midpoint in side $B C$ to form Figure 2. Figure 1 was then translated to the right along $\overset{―}{A C}$ so that $A^{″}$ coincides with $C$ to form Figure 3.

1.

Use Figure 3 to explain how you know the exterior angle $∠ B C C^{″}$ is equal to the sum of the $2$ remote interior angles $∠ B A C$ and $∠ A B C$ .

2.

Use Figure 3 to explain how you know the sum of the angles in a triangle is always $180 °$ .

3.

Use Figure 2 to explain how you know the sum of the angles in a quadrilateral is always $360 °$ .

4.

Use Figure 2 to explain how you know the opposite angles in a parallelogram are congruent.

5.

Use Figure 2 to explain how you know the opposite sides in a parallelogram are parallel and congruent.

6.

Use Figure 2 to explain how you know when two parallel lines are crossed by a transversal, the alternate interior angles are congruent.

7.

Use Figure 2 and/or 3 to explain how you know when two parallel lines are crossed by a transversal, the same-side interior angles are supplementary.

Set

Use the diagram for problems 8–10.

8.

Prove that $\overset{―}{C D}$ is an altitude of $△ A B C$ . Given $⨀ A ≅ ⨀ B$

Write a two-column proof.

9.

Prove that $△ A B C$ is an isosceles triangle. (You can choose the format of the proof you would like—paragraph, two-column, flow, etc.)

Given $⨀ A ≅ ⨀ B$

10.

Prove that $m ∠ A ≅ m ∠ B$ . (You can choose the proof format.)

11.

Construct the inscribed circle for the triangle.

12.

Construct the circumscribed circle for the triangle.

13.

Why is the intersection of the perpendicular bisectors of the three sides of a triangle the circumcenter of the triangle? In other words, why is the point of intersection of these segments the center of the circle that circumscribes the triangle? Construct the perpendicular bisector for the segment $\overset{―}{A B}$ . Pick any point $C$ on the perpendicular bisector, and connect it to points $A$ and $B$ . What is true about $\overset{―}{A C}$ and $\overset{―}{B C}$ ? Will this be true no matter where you place point $C$ ?

Go

Use what you know about triangles and parallelograms to find each measure.

14.

$Y X$

15.

$m ∠ Y X Z$

16.

$Y W$

17.

$L G$

18.

$H F$

19.

$m ∠ E H G$

20.

$m ∠ F E H$

21.

$m ∠ E L F$

22.

$F G$

23.

$E G$

24.

$m ∠ F G E$