Podcast
Questions and Answers
If angle PQR and angle XYZ are supplementary, and angle PQR is congruent to angle ABC, what can be concluded about angle XYZ and angle ABC?
If angle PQR and angle XYZ are supplementary, and angle PQR is congruent to angle ABC, what can be concluded about angle XYZ and angle ABC?
- Angle XYZ is congruent to angle ABC. (correct)
- Angle XYZ is complementary to angle ABC.
- Angle XYZ is supplementary to angle ABC.
- Angle XYZ is smaller than angle ABC.
If two lines are intersected by a transversal, and the same-side exterior angles are congruent, then the lines are parallel.
If two lines are intersected by a transversal, and the same-side exterior angles are congruent, then the lines are parallel.
False (B)
State the part-whole postulate for line segments.
State the part-whole postulate for line segments.
The sum of the parts is equal to the whole.
If lines a and b are both parallel to line c, then lines a and b are ________ to each other according to the transitivity of parallel lines.
If lines a and b are both parallel to line c, then lines a and b are ________ to each other according to the transitivity of parallel lines.
Match the angle relationships with their results when a transversal intersects two parallel lines:
Match the angle relationships with their results when a transversal intersects two parallel lines:
In triangle ABC, angle A measures 60 degrees and angle B measures 80 degrees. What is the measure of the exterior angle at vertex C?
In triangle ABC, angle A measures 60 degrees and angle B measures 80 degrees. What is the measure of the exterior angle at vertex C?
Given two triangles, if all three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are congruent.
Given two triangles, if all three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are congruent.
Which criterion can be used to prove that two triangles are congruent if all three sides of one triangle are congruent to the corresponding sides of the other triangle?
Which criterion can be used to prove that two triangles are congruent if all three sides of one triangle are congruent to the corresponding sides of the other triangle?
Which of the following statements is not a direct consequence of the Base Angles Theorem and its converse?
Which of the following statements is not a direct consequence of the Base Angles Theorem and its converse?
In any triangle, if an angle bisector is also an altitude, then the triangle must be equilateral.
In any triangle, if an angle bisector is also an altitude, then the triangle must be equilateral.
What is the measure of each angle in an equilateral triangle?
What is the measure of each angle in an equilateral triangle?
The intersection of the angle bisectors of a triangle is known as the ________.
The intersection of the angle bisectors of a triangle is known as the ________.
Match the point of concurrency in a triangle with its definition:
Match the point of concurrency in a triangle with its definition:
Which of the following conditions cannot guarantee that a triangle is isosceles?
Which of the following conditions cannot guarantee that a triangle is isosceles?
According to Side-Angle-Side (SAS), if two sides and any angle of one triangle are congruent to the corresponding sides and angle of another triangle, then the triangles are congruent.
According to Side-Angle-Side (SAS), if two sides and any angle of one triangle are congruent to the corresponding sides and angle of another triangle, then the triangles are congruent.
A right triangle is inscribed in a circle such that the hypotenuse coincides with the diameter of the circle. If the diameter of the circle is 10 cm, what is the length of the median to the hypotenuse?
A right triangle is inscribed in a circle such that the hypotenuse coincides with the diameter of the circle. If the diameter of the circle is 10 cm, what is the length of the median to the hypotenuse?
Flashcards
Part-Whole Postulate
Part-Whole Postulate
The sum of the measures of the parts equals the measure of the whole.
Supplementary Angles
Supplementary Angles
Angles that add up to 180 degrees are supplementary
Congruent Supplements Theorem
Congruent Supplements Theorem
Angles supplementary to congruent angles are congruent.
Vertical Angles
Vertical Angles
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Alternate Interior Angles
Alternate Interior Angles
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Same-Side Interior Angles
Same-Side Interior Angles
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Transitivity of Parallel Lines
Transitivity of Parallel Lines
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Triangle Angle Sum
Triangle Angle Sum
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SAS (Side, Angle, Side)
SAS (Side, Angle, Side)
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ASA (Angle, Side, Angle)
ASA (Angle, Side, Angle)
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Base Angles Theorem
Base Angles Theorem
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Base Angles Converse
Base Angles Converse
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Angles in Equilateral Triangle
Angles in Equilateral Triangle
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Angle Bisector to Base (Isosceles)
Angle Bisector to Base (Isosceles)
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Radius to Point of Tangency
Radius to Point of Tangency
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Tangent Segments
Tangent Segments
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Study Notes
- The sum of the parts equals the whole in geometry.
Angles
- The Parts-Whole Postulate states the sum of the parts equals the whole.
- If point B is in the interior of angle AOC, then m∠AOB + m∠BOC = m∠AOC.
- If m∠AOC is a straight angle, then m∠AOB + m∠BOC = 180 degrees.
- Angles supplementary to congruent angles are congruent.
- Vertical angles are congruent.
Parallel Lines
- Alternate interior/exterior angles are congruent if two lines are intersected by a transversal, then the lines are parallel.
- Same side interior/exterior angles are supplementary if 2 lines are intersected by a transversal, then the lines are parallel.
- If parallel lines are intersected by a transversal, then alternate interior and alternate exterior angles are congruent.
- If parallel lines are intersected by a transversal, then same side interior and same side exterior angles are supplementary.
- Transitivity of Parallel Lines: If two lines are parallel to a third line, then the two lines are parallel to each other in the same plane.
- If a line intersects one of two parallel lines in the same plane, it intersects the other line as well.
- In the same plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other line.
Angles in a Triangle
- Angle Sum: The sum of the measures of the angles in a triangle is 180 degrees.
- Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two remote interior angles.
- Exterior Angle Inequality: An exterior angle of a triangle is greater than either of its remote interior angles.
Congruent Triangles
- SSS (Side, Side, Side): Two triangles are congruent if all three sides of one are congruent to the corresponding sides of the other.
- SAS (Side, Angle, Side): Two triangles are congruent if two sides and the included angle of one are congruent to the corresponding sides and angle of the other.
- ASA (Angle, Side, Angle): Two triangles are congruent if two angles and the included side of one are congruent to the corresponding angles and side of the other.
Isosceles Triangle
- Base Angles Theorem: The base angles of an isosceles triangle are congruent.
- Base Angles Converse: If two angles in a triangle are congruent, then the sides opposite those angles are congruent, making the triangle isosceles.
- Angles in an Equilateral Triangle: The angles of an equilateral triangle are congruent, each measuring 60 degrees.
Angle Bisectors, Altitudes, and Medians in Isosceles Triangles
- Angle Bisector to Base: In an isosceles triangle, the angle bisector of the vertex angle is also the median and altitude to the base.
- Median to Base: The median to the base of an isosceles triangle is also an altitude and angle bisector of the vertex angle.
- Altitude to Base: The altitude to the base of an isosceles triangle is also the median and angle bisector of the vertex angle.
- Median is also the Altitude: If a median of a triangle is also an altitude, then the triangle is isosceles.
- Angle bisector is also the Altitude: If an angle bisector of a triangle is also an altitude, then the triangle is isosceles.
- Angle bisector is also the Median: If an angle bisector of a triangle is also its median, then the triangle is isosceles.
- Incenter is the intersection of angle bisectors.
- Orthocenter is the intersection of altitudes.
- Centroid is the intersection of medians.
Circles, Angles, and Tangents
- Radius to the Point of Tangency: If a line is tangent to a circle, then the radius to the point of tangency is perpendicular to that line.
- Tangent Segments: Tangent segments to a circle from the same point are congruent.
- Triangle inscribed in semi-circle, with one side as a diameter, is a right triangle.
- Conversely, if a right triangle is inscribed in a circle, it is inscribed in a semicircle, with the hypotenuse as a diameter.
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