### All Basic Geometry Resources

## Example Questions

### Example Question #1 : How To Find An Angle Of A Line

Examine the diagram. Which of these conditions does *not* prove that ?

**Possible Answers:**

Any of these statements can be used to prove that .

and

**Correct answer:**

If and , then , since two lines parallel to the same line are parallel to each other.

If , then , since two same-side interior angles formed by transversal are supplementary.

If , then , since two alternate interior angles formed by transversal are congruent.

However, regardless of whether and are parallel; they are vertical angles, and by the Vertical Angles Theorem, they *must* be congruent.

### Example Question #1 : How To Find An Angle Of A Line

An isosceles triangle has an interior angle that measures . What are the measures of its other two angles?

**Possible Answers:**

This triangle cannot exist.

**Correct answer:**

By the Isosceles Triangle Theorem, two interior angles must be congruent. However, since a triangle cannot have two obtuse interior angles, the two missing angles must be the ones that are congruent. Since the total angle measure of a triangle is , each of the missing angles measures .

### Example Question #2 : How To Find An Angle Of A Line

**Possible Answers:**

Straight

Acute

Scalene

Right

Obtuse

**Correct answer:**

Obtuse

Obtuse angles are greater than .

Scalene is a designation for triangles that have one angle greater than , but this figure is not a triangle.

Acute angles are less than , right angles are , and straight angles are .

Therefore this angle is obtuse.

### Example Question #2 : How To Find An Angle Of A Line

**Possible Answers:**

**Correct answer:**

When two parallel lines are crossed by a third line (called the transversal), the measure of the angles follows a specific pattern. The pairs of angles inside the two lines and on opposite sides are called alternate interior angles. Alternate interior angles, such as and , have the same degree measure. Therefore, the measure of is .

### Example Question #1 : How To Find An Angle Of A Line

Mark is training for cross country and comes across a new hill to run on. After Mark runs meters, he's at a height of meters. What is the hill's angle of depression when he's at an altitude of meters?

**Possible Answers:**

The same as the angle of inclination

Cannot be determined

**Correct answer:**

Upon reading the question, we're left with this spatial image of Mark in our heads. After adding in the given information, the image becomes more like

The hill Mark is running on can be seen in terms of a right triangle. This problem quickly becomes one that is asking for a mystery angle given that the two legs of the triangle are given. In order to solve for the angle of depression, we have to call upon the principles of the tangent function. Tan, Sin, or Cos are normally used when there is an angle present and the goal is to calculate one of the sides of the triangle. In this case, the circumstances are reversed.

Remember back to "SOH CAH TOA." In this problem, no information is given about the hypotenuse and nor are we trying to calculate the hypotenuse. Therefore, we are left with "TOA." If we were to check, this would work out because the angle at Mark's feet has the information for the opposite side and adjacent side.

Because there's no angle given, we must use the principles behind the tan function while using a fraction composed of the given sides. This problem will be solved using arctan (sometimes denoted as ).

### Example Question #1 : How To Find An Angle Of A Line

Two angles are supplementary and have a ratio of 1:4. What is the size of the smaller angle?

**Possible Answers:**

**Correct answer:**

Since the angles are supplementary, their sum is 180 degrees. Because they are in a ratio of 1:4, the following expression could be written:

### Example Question #1 : Plane Geometry

AB and CD are two parrellel lines intersected by line EF. If the measure of angle 1 is , what is the measure of angle 2?

**Possible Answers:**

**Correct answer:**

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.

### Example Question #12 : Plane Geometry

Lines A and B in the diagram below are parallel. The triangle at the bottom of the figure is an isosceles triangle.

What is the degree measure of angle ?

**Possible Answers:**

**Correct answer:**

Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles, and which will sum up to . Setting up an algebraic equation for this, we get . Solving for , we get . With this, we can get either (for the smaller angle) or (for the larger angle - must then use supplementary rule again for inner smaller angle). Either way, we find that the inner angles at the top are 80 degrees each. Since the sum of the angles within a triangle must equal 180, we can set up the equation as

degrees.

### Example Question #1 : Plane Geometry

In the figure above, lines AB and CD are parallel. Also, a, b, c, d, e, f, g, and h represent the measures of the angles in which they are shown. All of the following must be true EXCEPT:

**Possible Answers:**

b – h = e – c

f + d = a + g

a + h = e + d

f = b

180 – a – c = e + g – 180

**Correct answer:**

b – h = e – c

Lines AB and CD are parallel, so that means that d and e, which are in alternate interior angles, must be equal. Because a and d are vertical angles, and because e and h are vertical angles, this means that a = d and e = h. Therefore, a = d = e = h. Similarly, b = c = f = g.

Let us look at the choice f = b. We can see that this is true because b, c, f, and g are all equal.

We can then look at f + d = a + g. Let us subtract g from both sides, and then let us subtract d from both sides. This would give us the following equation.

f – g = a – d

Because f and g are equal, f – g = 0. Also, because a and d are the same, a – d = 0. Therefore, f – g = a – d = 0, so this is always true.

Now, let's look at 180 – a – c = e + g - 180. We can rewrite this as:

180 – (a + c) = e + g – 180.

a and c are supplementary, so a + c = 180. Likewise, e + g = 180. We can substitute 180 in for a + c and for e + g.

180 – 180 = 180 – 180 = 0

This means that 180 – a – c = e + g – 180 is indeed always true.

Next, let's examine the choice a + h = e + d. Let us subtract e from both sides and h from both sides. This will give us the following:

a – e = d – h

Because a = e = d = h, we could replace all of the values with a.

a – a = a – a = 0, so this is always true.

The final choice is b – h = e – c. Let us substitute c in for b and h in for e.

c – h = h – c

Let us add c and h to both sides.

2c = 2h

This means that c must equal h for this to be true. However, c does not always have to equal h. We know that c must equal f, and we know that f + h must equal 180. This means that c + h must equal 180. But this doesn't necessarily mean that c must equal h. In fact, this will only be true if c and h are both 90. Therefore, b – h = e – c isn't always true.

The answer is b – h = e – c.

### Example Question #1 : Act Math

Figure not drawn to scale.

In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?

**Possible Answers:**

50

66

114

40

33

**Correct answer:**

66

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.

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