Question

8 in △fgh, m∠f = m∠h, gf=x + 40, hf = 3x - 20 and gh = 2x+20. the length of gh is 1) 20 2) 40 3) 60 4) 80 9 the vertex - angle of an isosceles triangle measures 15 degrees more than one of its base angles. how many degrees are there in a base angle of the triangle? 1) 50 2) 55 3) 65 4) 70 10 in the diagram below of △aed and abcd, ae≅de. which statement is always true? 1) eb≅ec 2) ac≅db 3) ∠eba≅∠ecd 4) ∠eac≅∠edb 11 in △abc, ab≅bc. an altitude is drawn from b to ac and intersects ac at d. which conclusion is not always true? 1) ∠abd≅∠cbd 2) ∠bda≅∠bdc 3) ad≅bd 4) ad≅dc 12 in isosceles triangle abc, ab = bc. which statement will always be true? 1) m∠b = m∠a 2) m∠a>m∠b 3) m∠a = m∠c 4) m∠c<m∠b 13 if the vertex angles of two isosceles triangles are congruent, then the triangles must be 1) acute 2) congruent 3) right 4) similar 14 in isosceles triangle dog, the measure of the vertex angle is three - times the measure of one of the base angles. which statement about △dog is true? 1) △dog is a scalene triangle. 2) △dog is an acute triangle. 3) △dog is a right triangle. 4) △dog is an obtuse triangle.

Explanation:

Step1: Analyze question 9

Let the measure of a base - angle be \(x\) degrees. The vertex - angle is \(x + 15\) degrees. In a triangle, the sum of interior angles is \(180^{\circ}\). So, \(x+x+(x + 15)=180\).

Step2: Solve the equation

Combining like terms gives \(3x+15 = 180\). Subtract 15 from both sides: \(3x=180 - 15=165\). Then divide both sides by 3: \(x = 55\).

Step3: Analyze question 10

Since \(AE\cong DE\), \(\triangle AED\) is isosceles. \(\angle EAC\) and \(\angle EDB\) are angles formed by the non - congruent sides of \(\triangle AED\) with the line \(AD\). By the isosceles - triangle property and the fact that \(\triangle AED\) is isosceles, \(\angle EAC\cong\angle EDB\).

Step4: Analyze question 11

In an isosceles triangle \(\triangle ABC\) with \(AB = BC\) and \(BD\perp AC\), \(\triangle ABD\cong\triangle CBD\) (by HL congruence if considering right - triangles, or by SSS/SAS congruence). So, \(\angle ABD\cong\angle CBD\), \(\angle BDA\cong\angle BDC\), and \(AD\cong DC\). But \(AD\) is not always congruent to \(BD\).

Step5: Analyze question 12

In isosceles triangle \(ABC\) with \(AB = BC\), the base - angles are equal. So, \(\angle A=\angle C\).

Step6: Analyze question 13

If the vertex angles of two isosceles triangles are congruent, then the base - angles of the two isosceles triangles are also congruent (since in an isosceles triangle, if the vertex angle is \(v\), the base - angle is \(\frac{180 - v}{2}\)). By the AA (angle - angle) similarity criterion, the two triangles are similar.

Step7: Analyze question 14

Let the measure of a base - angle of \(\triangle DOG\) be \(x\) degrees. The vertex - angle is \(3x\) degrees. Then \(x + x+3x=180\), \(5x = 180\), \(x = 36\), and the vertex - angle is \(3x=108^{\circ}\). So, \(\triangle DOG\) is an obtuse triangle.

Answer:

1. 2) 55
2. 4) \(\angle EAC\cong\angle EDB\)
3. 3) \(\overline{AD}\cong\overline{BD}\)
4. 3) \(m\angle A=m\angle C\)
5. 4) similar
6. 4) \(\triangle DOG\) is an obtuse triangle.