Definitions
What is definition of congruent?
Two geometric figures are congruent if their measures are equal. (Measures of sides / Angle Measures)
Refer to your problem sheet. What triangle congruence theorem would prove that Triangle ABC is congruent to Triangle XYZ?
Side-Side-Side
State what triangle congruence theorems do NOT exist.
Angle-Angle-Angle and Angle-Side-Side
State which of the following is NOT a rigid transformation: Rotation, Dilation, Rotation, Translation.
Dilation because it does not create a shape that is congruent to its pre-image.
True or False: The definition of a parallelogram states that opposite sides are congruent.
FALSE
What is the definition of a segment bisector?
A line that intersects a segment at its midpoint to split the segment in half.
Refer to your problem sheet. What triangle congruence theorem would prove that Triangle ABC is congruent to Triangle XYZ?
What theorems can we use to prove that two lines are parallel?
Converse of Corresponding Angles Theorem or Converse of Interior Angles Theorem.
Jacob is using rigid transformations to prove that two figures (ABC and JKL) are congruent. Identify what TWO mistakes he has made in his proof.
1) Translate ABC by vector CK.
2) Rotate A'B'C' clockwise.
3) The figures line up so they are congruent.
1) Did not name a center of rotation or angle.
2) His congruence statement did not name the figures.
Refer to your problem sheet. HIKJ is a parallelogram. State the pairs of parallel lines.
HJ||IK and HI||JK
What is a rigid transformation? Give examples of some rigid transformations.
A transformation that produces a congruent image to some pre-image. Types of rigid transformations are reflection, rotation, and translation.
What triangle congruence theorems are possible given two pairs of congruent angles and one pair of congruent side lengths?
Angle-Side-Angle or Angle-Angle-Side
What statement can we make about corresponding angles in two congruent figures?
They are congruent based on the Corresponding Parts of Congruent Figures Theorem
Why do rigid transformations allow us to prove congruence?
Rigid transformations preserve size, shape, and angle measures, thus producing a congruent image to some pre-image.
Refer to your problem sheet. ABCD is a parallelogram. State the two pairs of congruent alternate interior angles.
Angle DAC ≅ Angle BCA; Angle CAB ≅ Angle ACD
What is a parallelogram?
A quadrilateral with opposite pairs of parallel sides.
Refer to your problem sheet. Is Triangle ABC congruent to Triangle DEF?
Not enough information since we are only given Angle-Side-Side
What condition is needed for the Alternate Interior Angles theorem to be true?
Two parallel lines cut by a transversal
Refer to your problem sheet. Use rigid transformations to prove if triangle ABC is congruent to triangle YXZ.
Triangle ABC is NOT congruent to Triangle YXZ. The corresponding points do not map to each other.
Refer to your problem sheet. HIKJ is a parallelogram. Prove that triangle JHI is congruent to triangle IKJ.
Given HIJK is a parallelogram, we know that line HI||JK and line HJ||IK. Thus by the Alternate Interior Angles Theorem, angle HIJ ≅ angle KJI and angle HJI ≅ angle KIJ. Additionally by the reflexive property, line JI ≅ line IJ. Thus by the Angle-Side-Angle Triangle Congruence Theorem, triangle JHI is congruent to triangle IKJ.
What is an isosceles triangle?
A triangle with two congruent sides.
Refer to your problem sheet. ABDC is a parallelogram. What triangle congruence theorem can be used to prove that Triangle ACD is congruent to Triangle CAB?
Angle-Side-Angle
What theorem could we use to solve for missing angles formed by two intersecting lines?
Vertical Angle theorem and Straight Angle theorem
Refer to your reference sheet. Use rigid transformations to justify that triangle ABC is congruent to triangle DEF.
- Translate triangle ABC by vector AD.
- Reflect A'B'C' over line DE.
- Triangles ABC and DEF are congruent.
Refer to your problem sheet. ACEG is a parallelogram where lines EC and BF bisect each other. Prove that line BC is congruent to line FE.
- Given that lines EC and BF bisect each other, then line BD ≅ line FD and line ED ≅ line CD by the definition of a bisector.
- Angle EDF ≅ angle CDB by the Vertical Angles Theorem.
- Triangles EDF ≅ triangle CDB by the Side-Angle-Side Triangle Congruence Theorem.
- Line BC ≅ line FE as they are corresponding part of the two congruent triangles.