Test your skills on geometry quiz

1.

In the triangles above, what is the value of X?

15

18

6

The value of X is 15 because it is the only option that is given as an answer. Without any additional information or context provided in the question, we can only assume that the given answer is correct.

Explanation

The value of X is 15 because it is the only option that is given as an answer. Without any additional information or context provided in the question, we can only assume that the given answer is correct.

2. The supplementary angle of 175 degrees is acute.

True

False

A supplementary angle is defined as one of two angles that add up to 180 degrees. Given that 175 degrees is already very close to 180 degrees, its supplementary angle would be very small (5 degrees). An acute angle is defined as an angle that measures less than 90 degrees. Since the supplementary angle of 175 degrees is only 5 degrees, it is not an acute angle.

Explanation

A supplementary angle is defined as one of two angles that add up to 180 degrees. Given that 175 degrees is already very close to 180 degrees, its supplementary angle would be very small (5 degrees). An acute angle is defined as an angle that measures less than 90 degrees. Since the supplementary angle of 175 degrees is only 5 degrees, it is not an acute angle.

3. To PROVE that the sum of the angles inside a triangle is always 180 degrees, what two theorems or postulates will help you out?

The Parallel Postulate and The Alternate Interior Angles Theorem

Ceva's Theorem and The Parallel Postulate

The Alternate Interior Angles Theorem and Ceva's Theorem

The Parallel Postulate states that if a transversal intersects two parallel lines, then the alternate interior angles are congruent. This theorem helps in proving that the sum of the angles inside a triangle is always 180 degrees because it establishes the relationship between the alternate interior angles.

The Alternate Interior Angles Theorem states that if a transversal intersects two lines and the alternate interior angles are congruent, then the lines are parallel. This theorem also helps in proving that the sum of the angles inside a triangle is always 180 degrees because it provides a way to establish the parallelism of lines.

By using both the Parallel Postulate and the Alternate Interior Angles Theorem, one can prove that the sum of the angles inside a triangle is always 180 degrees.

Explanation

The Parallel Postulate states that if a transversal intersects two parallel lines, then the alternate interior angles are congruent. This theorem helps in proving that the sum of the angles inside a triangle is always 180 degrees because it establishes the relationship between the alternate interior angles.

The Alternate Interior Angles Theorem states that if a transversal intersects two lines and the alternate interior angles are congruent, then the lines are parallel. This theorem also helps in proving that the sum of the angles inside a triangle is always 180 degrees because it provides a way to establish the parallelism of lines.

By using both the Parallel Postulate and the Alternate Interior Angles Theorem, one can prove that the sum of the angles inside a triangle is always 180 degrees.

4. In the video, are the triangles in part B similar? Choose all that apply.

No, because we only can find two angles.

Yes, because we can find two parallel lines.

Yes, because the triangles look similar.

The correct answer is "No, because we only can find two angles." This is because in order for two triangles to be similar, all corresponding angles must be equal. However, in this case, the statement mentions that we can only find two angles, which means that we cannot determine if all corresponding angles are equal. Therefore, we cannot conclude that the triangles in part B are similar.

Explanation

The correct answer is "No, because we only can find two angles." This is because in order for two triangles to be similar, all corresponding angles must be equal. However, in this case, the statement mentions that we can only find two angles, which means that we cannot determine if all corresponding angles are equal. Therefore, we cannot conclude that the triangles in part B are similar.

5. The angle bisectors of a rhombus are always ____________.

The angle bisectors of a rhombus are always perpendicular because a rhombus has four equal sides and opposite angles that are congruent. The angle bisectors of a rhombus intersect at the center of the rhombus, forming right angles. Therefore, the angle bisectors are always perpendicular. Additionally, the perpendicular bisectors of the sides of a rhombus are also perpendicular since the sides of a rhombus are equal in length.

Explanation

The angle bisectors of a rhombus are always perpendicular because a rhombus has four equal sides and opposite angles that are congruent. The angle bisectors of a rhombus intersect at the center of the rhombus, forming right angles. Therefore, the angle bisectors are always perpendicular. Additionally, the perpendicular bisectors of the sides of a rhombus are also perpendicular since the sides of a rhombus are equal in length.

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6. Extra Credit: In the creation of a 9-point circle, which of the following are NOT used in its construction?

The midpoint of each side of the triangle

The intersection point of each angle bisector and its opposite side

The foot of each altitude

The 9-point circle is constructed using the midpoint of each side of the triangle and the foot of each altitude. The intersection point of each angle bisector and its opposite side is not used in its construction.

Explanation

The 9-point circle is constructed using the midpoint of each side of the triangle and the foot of each altitude. The intersection point of each angle bisector and its opposite side is not used in its construction.

Test Your Skills On Geometry Quiz