Question

i can apply knowledge of triangles.

1. use the triangle at the right to answer the questions.

a. angle xyz is opposite side ______
b. angle zxy is opposite side ______
c. angle yzx is opposite side ______

1. find the missing angle measure in each triangle below.

1. three students wrote side lengths that they thought would form a triangle. which student(s) were correct? justify your response.

keisha 10 cm, 15 cm, 24 cm
miguel 21 cm, 7 cm, 6 cm
ralphie 9 cm, 22 cm, 11 cm

i can construct triangles.

1. marcella draws the line segment shown below to construct a triangle. finish the construction so that m∠a = 55° and m∠b = 35°. what is the measure of ∠acb? ______
2. construct triangle xyz, where the m∠yxz = 30°, the m∠xyz = 70°, and the length of (overline{xy}) is 2 inches.

i can determine the conditions for a unique triangle, more than one triangle, or no triangle.

1. the cards below contain clues. determine whether the conditions will result in one unique triangle, more than one triangle, or no triangle. justify your solution with a sketch and description below.

a (\triangle abc) ab is 6.3 cm, bc is 4.2 cm, ca is 10.4 cm
b (\triangle def) de is 2 cm, ef is 3 cm, fd is 5 cm
c (\triangle ghi) ∠g is 80°, ∠h is 42°, ∠i is 58°

Explanation:

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Problem 13

Step1: Identify opposite side (∠XYZ)

In $\triangle XYZ$, $\angle XYZ$ (at $Y$) is opposite side $XZ$.

Step2: Identify opposite side (∠ZXY)

In $\triangle XYZ$, $\angle ZXY$ (at $X$) is opposite side $YZ$.

Step3: Identify opposite side (∠YZX)

In $\triangle XYZ$, $\angle YZX$ (at $Z$) is opposite side $XY$.

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Problem 14

Step1: Calculate missing ∠ for △LMN

Sum of angles = $180^\circ$. Missing angle: $180 - 88 - 33 = 59^\circ$

Step2: Calculate missing ∠ for △QRS

Sum of angles = $180^\circ$. Missing angle: $180 - 29.4 - 74 = 76.6^\circ$

Step3: Calculate missing ∠ for △XYZ

Sum of angles = $180^\circ$. Missing angle: $180 - 40 - 40 = 100^\circ$

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Problem 15

Step1: Check Keisha's sides (10,15,24)

Triangle Inequality: $10+15>24$ ($25>24$), $10+24>15$, $15+24>10$. All true.

Step2: Check Miguel's sides (21,7,6)

Triangle Inequality: $7+6=13 < 21$. Fails condition.

Step3: Check Ralphie's sides (9,22,11)

Triangle Inequality: $9+11=20 < 22$. Fails condition.

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Problem 16

Step1: Sum of angles in △ABC

Sum of angles = $180^\circ$. $\angle ACB = 180 - 55 - 35 = 90^\circ$

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Problem 17

Step1: Draw segment $\overline{XY}=2$ inches

Use ruler to make 2-inch $\overline{XY}$.

Step2: Construct $\angle XYZ=70^\circ$ at $Y$

Use protractor to draw $70^\circ$ angle from $Y$.

Step3: Construct $\angle YXZ=30^\circ$ at $X$

Use protractor to draw $30^\circ$ angle from $X$.

Step4: Mark intersection as $Z$

The two angle rays meet at $Z$; $\triangle XYZ$ is complete.

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Problem 18

Step1: Check △ABC (6.3,4.2,10.4)

Triangle Inequality: $6.3+4.2=10.5 \approx 10.4$, $10.5$ is not greater than $10.4$. No triangle.

Step2: Check △DEF (2,3,5)

Triangle Inequality: $2+3=5$, not greater than 5. No triangle.

Step3: Check △GHI (80°,42°,58°)

Sum of angles = $180^\circ$. Given all 3 angles, infinite similar triangles, but if side lengths are not fixed, but with angle measures fixed, it forms one unique triangle shape (unique up to similarity; if no side constraint, but standardly, fixed angles give unique triangle type).

Answer:

1. a. $XZ$

b. $YZ$
c. $XY$

1. A. $59^\circ$

B. $76.6^\circ$
C. $100^\circ$

1. Keisha
2. $90^\circ$
3. (See explanation for construction steps)
4. A. No triangle

B. No triangle
C. One unique triangle