Lesson Summary - 7.5 PDF

Preview

Summary

lesson summary...

Description

Colin O’Brien 1/17/18 Lesson Summary: 7.5 - Congruent Triangles to the Rescue Concepts We Covered: - Proving that isosceles triangles have a line of symmetry. For example, we proved that Zac’s claim is true that all isosceles triangles have a line of symmetry that goes through the vertex angle and the midpoint of the base. We know this because since the line goes through the vertex angle AND the midpoint of the base the line is perpendicular to the base because, in the model triangle, AC = BC so they meet equidistant from each other which is the midpoint of AB so they are perpendicular. So the line of reflection, CD, creates right triangles ADC and BDC. We know these are congruent because we know they share congruent hypotenuses which are AC and BC and AD = BD so using the property HL, the right triangles are congruent and so CD is a line of symmetry. - What is told by the line of symmetry within an isosceles triangle. For example, using the line of symmetry, you make 2 right triangles as I described above. Since in congruent triangles corresponding parts are congruent, you know that A = B and ACD = BCD. Also, CD is a perpendicular bisector of the base because as we know above CD is perpendicular to AB and goes through its midpoint. CD is also an angular bisector of C because ACD = BCD so CD splits angle C into two congruent angles so by definition CD angularly bisects C. - Using congruent triangles to find other geometric properties of shapes. For example, we had to find information about a rhombus by using our knowledge of congruent triangles, and the definition of a rhombus which is that a rhombus has four congruent sides. Using the rhombus ABCD, we figured out that the diagonals bisect each other. To prove this we first figured out that DNA = BNA. We figured this out using HL because we already know that DA = BA based on the definition of a rhombus, and we

know that AC = AC so we have a congruent hypotenuse and leg. We know that DDDNA and BNA are right triangles because the diagonal AC is a line of symmetry for the isosceles triangle DAB because it goes through the vertex angle and the midpoint of the base. This means that angles 1 and 2 are right angles making right triangles making DNA = BNA. Therefore, since corresponding parts of congruent triangles are congruent, DN = BN so AC is a perpendicular bisector of DB. We know that DB is a perpendicular bisector of AC because DNC = DNA. We know this because DB is a line of symmetry for the isosceles triangle DAC because DB goes through the midpoint of the base and the vertex angle. So, DC = DA and DN = DN so using HL we know that DNC = DNA. Since corresponding parts of congruent triangles are congruent, CN = AN so DB is a perpendicular bisector of AC. Vocabulary We Learned: - Isosceles Triangle: An isosceles triangle is a triangle with at least 2 congruent sides. For example, ABC is an isosceles triangle because AC = BC - Leg: One of the congruent sides of an isosceles triangle. For example, in triangle ABC, AC and BC are legs. - Base: The non congruent side of an isosceles triangle. For example, in triangle ABC, AB is the base. - Vertex angle: The angle formed by the legs. For example, angle of ABC - Base angles: The angles next to the base. For example, angles for ABC

C is a vertex

A and

B are base

- Inductive Reasoning: Reasoning that is based on trends that are assumed to continue. For example, based on my triangles QWE, ASD, ZXC, and

DDFGH, we can assume that SSA guarantees congruence because based on the pattern so far, triangles that share the criteria SSA are congruent. - Deductive Reasoning: Reasoning that uses facts and hard evidence to prove something. For example, looking at triangles BNM and JKL we know that SSA does not guarantee congruence because you can see that the triangles are not congruent but share the criteria....