All ACT Math Resources
Example Questions
Example Question #221 : Geometry
Isosceles triangles and share common side . is an obtuse triangle with sides . is also an obtuse isosceles triangle, where . What is the measure of ?
In order to prove triangle congruence, the triangles must have SAS, SSS, AAS, or ASA congruence. Here, we have one common side (S), and no other demonstrated congruence. Hence, we cannot guarantee that side is not one of the two congruent sides of , so we cannot state congruence with .
Example Question #11 : Acute / Obtuse Isosceles Triangles
Points A and B lie on a circle centered at Z, where central angle <AZB measures 140°. What is the measure of angle <ZAB?
Cannot be determined from the given information
25°
30°
20°
15°
20°
Because line segments ZA and ZB are radii of the circle, they must have the same length. That makes triangle ABZ an isosceles triangle, with <ZAB and <ZBA having the same measure. Because the three angles of a triangle must sum to 180°, you can express this in the equation:
140 + 2x = 180 --> 2x = 40 --> x = 20
Example Question #11 : Acute / Obtuse Isosceles Triangles
Triangle FGH has equal lengths for FG and GH; what is the measure of ∠F, if ∠G measures 40 degrees?
40 degrees
100 degrees
None of the other answers
70 degrees
140 degrees
70 degrees
It's good to draw a diagram for this; we know that it's an isosceles triangle; remember that the angles of a triangle total 180 degrees.
Angle G for this triangle is the one angle that doesn't correspond to an equal side of the isosceles triangle (opposite side to the angle), so that means ∠F = ∠H, and that ∠F + ∠H + 40 = 180,
By substitution we find that ∠F * 2 = 140 and angle F = 70 degrees.
Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle
The vertex angle of an isosceles triangle is . What is the base angle?
An isosceles triangle has two congruent base angles and one vertex angle. Each triangle contains . Let = base angle, so the equation becomes . Solving for gives
Example Question #222 : Plane Geometry
In an isosceles triangle the base angle is five less than twice the vertex angle. What is the sum of the vertex angle and the base angle?
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let = the vertex angle
and = base angle
So the equation to solve becomes
or
Thus the vertex angle is 38 and the base angle is 71 and their sum is 109.
Example Question #1 : Isosceles Triangles
Sides and in this triangle are equal. What is the measure of ?
This triangle has an angle of . We also know it has another angle of at because the two sides are equal. Adding those two angles together gives us total. Since a triangle has total, we subtract 130 from 180 and get 50.
Example Question #3 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle
An isosceles triangle has a base angle that is six more than three times the vertex angle. What is the base angle?
Every triangle has 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.
Let = vertex angle and = base angle.
Then the equation to solve becomes
or
.
Solving for gives a vertex angle of 24 degrees and a base angle of 78 degrees.
Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle
The base angle of an isosceles triangle is thirteen more than three times the vertex angle. What is the difference between the vertex angle and the base angle?
Every triangle has . An isosceles triangle has one vertex ange, and two congruent base angles.
Let be the vertex angle and be the base angle.
The equation to solve becomes , since the base angle occurs twice.
Now we can solve for the vertex angle.
The difference between the vertex angle and the base angle is .
Example Question #5 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle
A particular acute isosceles triangle has an internal angle measuring . Which of the following must be the other two angles?
By definition, an acute isosceles triangle will have at least two sides (and at least two corresponding angles) that are congruent, and no angle will be greater than . Addtionally, like all triangles, the three angles will sum to . Thus, of our two answers which sum to , only is valid, as would violate the "acute" part of the definition.
Example Question #4 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle
In triangle ABC, Angle A = x degrees, Angle B = 2x degrees, and Angle C = 3x+30 degrees. How many degrees is Angle B?
45°
105°
30°
50°
25°
50°
Because the interior angles of a triangle add up to 180°, we can create an equation using the variables given in the problem: x+2x+(3x+30)=180. This simplifies to 6X+30=180. When we subtract 30 from both sides, we get 6x=150. Then, when we divide both sides by 6, we get x=25. Because Angle B=2x degrees, we multiply 25 times 2. Thus, Angle B is equal to 50°. If you got an answer of 25, you may have forgotten to multiply by 2. If you got 105, you may have found Angle C instead of Angle B.