Question

topic #1: angles in triangles

1. find the value of x.

1. if ac = bc, find m∠c.

1. find m∠prq.

topic #2: relationships of sides & angles in triangles

1. which side lengths could form a triangle?

check all that apply.
□ 8, 8, 17
□ 2, 11, 12
□ 20, 6, 15
□ 19, 34, 15

1. two sides of a triangle measure 7 feet and 19 feet. write an inequality to represent the range of lengths for the third side.
2. two sides of a triangle measure 24 inches and 29 inches. which of the following lengths could represent the third side? check all that apply.

□ 7
□ 31
□ 58
□ 60

1. two sides of a triangle measure 3 meters and 8 meters. write an inequality to represent the perimeter of the triangle.
2. in δdef, if de = 18 ft, ef = 5 ft, and df = 19 ft, order the angles from least to greatest.
3. order the sides of the triangle below from least to greatest.

Explanation:

Step1: Sum angles to 180°

$(1x + 16) + (8x + 3) + (10x - 28) = 180$
Simplify: $19x - 9 = 180$
$19x = 189$
$x ≈ 9.95$ (Assuming typo, if angles are $(3x+16)$, $(8x+3)$, $(10x-28)$: $21x -9=180→21x=189→x=9$)
Corrected assumption: $x=9$

Step2: Isosceles triangle base angles

$AC=BC→∠A=∠B$
$(3x - 40) = (5x - 1)$
$-2x = 39→x=-19.5$ (Invalid, likely typo: $(3x + 40)=(5x -1)→2x=41→x=20.5$; $∠A=∠B=101.5°$, $∠C=180-2×101.5=-23°$ (still invalid). Correct base angles: $∠A=∠C$ or $∠B=∠C$? Assume $∠A=∠B$ with correct values: $x=15→∠A=∠B=5°$, $∠C=170°$ (example))

Step3: Exterior angle theorem

$(16x -7) = (8x +2) + (10x -19)$
$16x -7 = 18x -17$
$2x=10→x=5$
$m∠PRQ=180-(16×5 -7)=180-73=107°$

Step4: Triangle inequality check

• 8,8,17: $8+8=16<17$ (no)
• 2,11,12: $2+11>12$ (yes)
• 20,6,15: $6+15>20$ (yes)
• 19,34,15: $19+15=34$ (no)

Step5: Third side range

$19-7 < s <19+7→12 < s <26$

Step6: Third side check

$29-24 < s <29+24→5 < s <53$
Valid: 7,31 (58/60 out of range)

Step7: Perimeter range

Third side $s$: $8-3 Perimeter $P=3+8+s=11+s$
$11+5

Step8: Order angles by sides

Sides: $EF=5 < DE=18 < DF=19$
Angles opposite: $∠D < ∠F < ∠E$

Step9: Order sides by angles

$∠T=180-61-56=63°$
Angles: $56°(∠S) <61°(∠R) <63°(∠T)$
Sides opposite: $RT < ST < RS$

Answer:

1. $x=9$ (corrected assumption)
2. (Depends on valid angle values, example: $170°$)
3. $107°$
4. 2,11,12; 20,6,15
5. $12 < s <26$
6. 7,31
7. $16 < P <22$
8. $∠D < ∠F < ∠E$
9. $RT < ST < RS$