Lesson Menu FiveMinute Check over Chapter 4 Mathematical

Lesson Menu Five-Minute Check (over Chapter 4) Mathematical Practices Then/Now New Vocabulary Theorems: Perpendicular Bisectors Example 1: Use the Perpendicular Bisector Theorems Theorem 5. 3: Circumcenter Theorem Proof: Circumcenter Theorem Example 2: Real-World Example: Use the Circumcenter Theorems: Angle Bisectors Example 3: Use the Angle Bisector Theorems Theorem 5. 6: Incenter Theorem Example 4: Use the Incenter Theorem

Over Chapter 4 Find x if m A = 10 x + 15, m B = 8 x – 18, and m C = 12 x + 3. A. 3. 75 B. 6 C. 12 D. 16. 5

Over Chapter 4 Name the corresponding congruent sides if ΔRST ΔUVW. A. R V, S W, T U B. R W, S U, T V C. R U, S V, T W D. R U, S W, T V

Over Chapter 4 Name the corresponding congruent sides if ΔLMN ΔOPQ. A. B. C. D. ,

Over Chapter 4 Find y if ΔDEF is an equilateral triangle and m F = 8 y + 4. A. 22 B. 10. 75 C. 7 D. 4. 5

Over Chapter 4 ΔABC has vertices A(– 5, 3) and B(4, 6). What are the coordinates for point C if ΔABC is an isosceles triangle with vertex angle A? A. (– 3, – 6) B. (4, 0) C. (– 2, 11) D. (4, – 3)

Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. Content Standards G. CO. 10 Prove theorems about triangles. G. MG. 3 Apply geometric methods to solve problems (e. g. , designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

You used segment and angle bisectors. • Identify and use perpendicular bisectors in triangles. • Identify and use angle bisectors in triangles.

• perpendicular bisector • concurrent lines • point of concurrency • circumcenter • incenter

Use the Perpendicular Bisector Theorems A. Find BC. BC = AC Perpendicular Bisector Theorem BC = 8. 5 Substitution Answer: 8. 5

Use the Perpendicular Bisector Theorems B. Find XY. Answer: 6

Use the Perpendicular Bisector Theorems C. Find PQ. PQ = RQ 3 x + 1 = 5 x – 3 Substitution 1 = 2 x – 3 Subtract 3 x from each side. 4 = 2 x Add 3 to each side. 2 =x Divide each side by 2. So, PQ = 3(2) + 1 = 7. Answer: 7 Perpendicular Bisector Theorem

A. Find NO. A. 4. 6 B. 9. 2 C. 18. 4 D. 36. 8

B. Find TU. A. 2 B. 4 C. 8 D. 16

C. Find EH. A. 8 B. 12 C. 16 D. 20

Use the Circumcenter Theorem GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? By the Circumcenter Theorem, a point equidistant from three points is found by using the perpendicular bisectors of the triangle formed by those points.

Use the Circumcenter Theorem Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle. C Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle.

BILLIARDS A triangle used to rack pool balls is shown. Would the circumcenter be found inside the triangle? A. No, the circumcenter of an acute triangle is found in the exterior of the triangle. B. Yes, circumcenter of an acute triangle is found in the interior of the triangle.

Use the Angle Bisector Theorems A. Find DB. DB = DC Angle Bisector Theorem DB = 5 Substitution Answer: DB = 5

Use the Angle Bisector Theorems B. Find m WYZ.

Use the Angle Bisector Theorems WYZ XYW Definition of angle bisector m WYZ = m XYW Definition of congruent angles m WYZ = 28 Substitution Answer: m WYZ = 28

Use the Angle Bisector Theorems C. Find QS. QS = SR 4 x – 1 = 3 x + 2 x– 1 =2 x =3 Angle Bisector Theorem Substitution Subtract 3 x from each side. Add 1 to each side. Answer: So, QS = 4(3) – 1 or 11.

A. Find the measure of SR. A. 22 B. 5. 5 C. 11 D. 2. 25

B. Find the measure of HFI. A. 28 B. 30 C. 15 D. 30

C. Find the measure of UV. A. 7 B. 14 C. 19 D. 25

Use the Incenter Theorem A. Find SU if S is the incenter of ΔMNP. By the Incenter Theorem, since S is equidistant from the sides of ΔMNP, ST = SU. Find SU by using the Pythagorean Theorem. a 2 + b 2 = c 2 Pythagorean Theorem 82 + SU 2 = 102 Substitution 64 + SU 2 = 100 82 = 64, 102 = 100

Use the Incenter Theorem SU 2 = 36 SU = ± 6 Subtract 64 from each side. Take the square root of each side. Since length cannot be negative, use only the positive square root, 6. Since ST = SU, SU = 6. Answer: SU = 6

Use the Incenter Theorem B. Find m SPU if S is the incenter of ΔMNP. Since MS bisects RMT, m RMT = 2 m RMS. So m RMT = 2(31) or 62. Likewise, m TNU = 2 m SNU, so m TNU = 2(28) or 56.

Use the Incenter Theorem m UPR + m RMT + m TNU = 180 m UPR + 62 + 56 = 180 m UPR + 118 = 180 m UPR = 62 Triangle Angle Sum Theorem Substitution Simplify. Subtract 118 from each side. Since PS bisects UPR, 2 m SPU = m UPR. This 1 m UPR. means that m SPU = __ 2 1 (62) or 31 Answer: m SPU = __ 2

A. Find the measure of GF if D is the incenter of ΔACF. A. 12 B. 144 C. 8 D. 65

B. Find the measure of BCD if D is the incenter of ΔACF. A. 58° B. 116° C. 52° D. 26°