Question

hw: types and properties of triangles

1. classify each triangle by its angles and sides.

a. △abe: right scalene
b. △bec: equiangk isosceles
c. △def: equiangk equilateral
d. △cdf: obtuse scalene

1. m∠1 = __ m∠2 = m∠3 = 64 m∠4 = m∠5 = __
2. m∠1 = __ m∠2 = m∠3 = m∠4 = m∠5 = m∠6 = m∠7 = __
3. x = ____
4. x = ____
5. x = ____
6. x = ____
7. m∠e = __ m∠g = __
8. △cde is an isosceles triangle with ∠d≅∠e. if cd = 4x + 9, de = 7x - 5, and ce = 16x - 27, find x and the measure of each side. x = __ cd = de = ce = __
9. write three valid congruency statements given the triangles below.

a) ____
b) ____
c) ____

1. if △kpl≅△acm, complete each part.

a) kl≅ __ d) ∠p≅ __
b) ac≅ __ e) ∠k≅ __
c) pl≅ __ f) ∠m≅ __

1. if △wxy≅△hji, complete each part.

a) ji = ____
b) jh = ____
c) m∠w = ____
d) m∠j = ____
e) m∠i = ____

1. determine if the triangles below are congruent. if yes, state which method.

a)
b)
c)
d)

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. For problem 2, in the first triangle, if one angle is 88° and another is 31°, then \(m\angle2=180-(88 + 31)=61^{\circ}\). In the second triangle of problem 2, if two angles are 64° and 51°, then \(m\angle4=180-(64 + 51)=65^{\circ}\), and \(m\angle3 = 180 - 61-65 = 54^{\circ}\), \(m\angle1=180 - 88=92^{\circ}\), \(m\angle5 = 180 - 64=116^{\circ}\).

Step2: Solve for \(x\) in angle - sum equations

For problem 4: \((3x - 4)+(9x + 17)+(6x-31)=180\). Combine like - terms: \(3x+9x + 6x-4 + 17-31 = 180\), \(18x-18 = 180\), \(18x=198\), \(x = 11\).
For problem 5: \((5x-27)+(10x - 23)+(7x-26)=180\). Combine like - terms: \(5x+10x + 7x-27-23-26 = 180\), \(22x-76 = 180\), \(22x=256\), \(x=\frac{128}{11}\approx11.64\).
For problem 6: \((2x + 20)+(11x-60)+(5x-11)=180\). Combine like - terms: \(2x+11x + 5x+20-60-11 = 180\), \(18x-51 = 180\), \(18x=231\), \(x=\frac{77}{6}\approx12.83\).

Step3: Solve for \(x\) in isosceles triangle

For problem 7, since it is an equilateral triangle, \(9x-3 = 60\), \(9x=63\), \(x = 7\).
For problem 8, in \(\triangle EFG\), \(m\angle E=\frac{180 - 106}{2}=37^{\circ}\), \(m\angle G = 37^{\circ}\).
For problem 9, since \(\triangle CDE\) is isosceles with \(\angle D\cong\angle E\), then \(CD = DE\). So \(4x + 9=7x-5\), \(7x-4x=9 + 5\), \(3x=14\), \(x=\frac{14}{3}\). Then \(CD=4x + 9=4\times\frac{14}{3}+9=\frac{56 + 27}{3}=\frac{83}{3}\), \(DE=\frac{83}{3}\), \(CE=16\times\frac{14}{3}-27=\frac{224-81}{3}=\frac{143}{3}\).

Step4: Write congruency statements

For problem 10: \(\triangle PQR\cong\triangle TSU\), \(\triangle QRP\cong\triangle SUT\), \(\triangle RPQ\cong\triangle UTS\).

Step5: Identify corresponding parts of congruent triangles

For problem 11: If \(\triangle KPL\cong\triangle ACM\), then \(\overline{KL}\cong\overline{CM}\), \(\overline{AC}\cong\overline{KP}\), \(\overline{PL}\cong\overline{AM}\), \(\angle P\cong\angle M\), \(\angle K\cong\angle A\), \(\angle M\cong\angle P\).
For problem 12: If \(\triangle WXY\cong\triangle HJI\), then \(JI = XY = 16m\), \(JH=WX = 15m\), \(m\angle W=m\angle H\), \(m\angle J=m\angle X = 114^{\circ}\), \(m\angle I=m\angle Y=37^{\circ}\).

Step6: Determine triangle congruency

For problem 13:
a) SSS (Side - Side - Side) if the marked sides are equal.
b) SAS (Side - Angle - Side) if the marked side - angle - side are equal.
c) ASA (Angle - Side - Angle) if the marked angle - side - angle are equal.
d) HL (Hypotenuse - Leg) for right - triangles if the hypotenuse and one leg are equal.

Answer:

2: \(m\angle1 = 92^{\circ}\), \(m\angle2=61^{\circ}\), \(m\angle3 = 54^{\circ}\), \(m\angle4=65^{\circ}\), \(m\angle5 = 116^{\circ}\)
4: \(x = 11\)
5: \(x=\frac{128}{11}\)
6: \(x=\frac{77}{6}\)
7: \(x = 7\)
8: \(m\angle E = 37^{\circ}\), \(m\angle G = 37^{\circ}\)
9: \(x=\frac{14}{3}\), \(CD=\frac{83}{3}\), \(DE=\frac{83}{3}\), \(CE=\frac{143}{3}\)
10: a) \(\triangle PQR\cong\triangle TSU\), b) \(\triangle QRP\cong\triangle SUT\), c) \(\triangle RPQ\cong\triangle UTS\)
11: a) \(\overline{KL}\cong\overline{CM}\), b) \(\overline{AC}\cong\overline{KP}\), c) \(\overline{PL}\cong\overline{AM}\), d) \(\angle P\cong\angle M\), e) \(\angle K\cong\angle A\), f) \(\angle M\cong\angle P\)
12: a) \(JI = 16m\), b) \(JH=15m\), c) \(m\angle W=m\angle H\), d) \(m\angle J = 114^{\circ}\), e) \(m\angle I=37^{\circ}\)
13: a) SSS, b) SAS, c) ASA, d) HL