Problem 1
Short Answers:
b) Definition of a Rhombus
c) Given
d) Definition of a Rhombus
e) Transitive Property of Equality
f) SSS Congruence Property
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Explanations:
b) Since RUMH is a rhombus, we know that the four sides of the figure are all the same length. This is simply what the definition of wha "rhombus" means. So RU = UM = MH = HR, which can be rearranged to state what statement b is showing.
c) This is given up at the top. All we do is simply repeat what we're given. It must be written exactly as stated.
d) Similar to the reason for (b). All sides of a rhombus are the same length.
e) If RH = RU (see statement b) and RU = MB (see statement d), then RH = MB by the transitive property of equality. Think of it as a chain. If we go from point A to point B to C, then we can shorten things and go straight from A to C (skipping B altogether). This idea is used multiple times to tie together all of the segments shown for this statement.
f) If we know that a triangle has 3 congruent sides, then it must be equilateral. We have two equilateral triangles RHM and UMB, both of which have sides that are congruent corresponding sides. So that's why they are congruent. Specifically I'm using the SSS (side side side) Congruence Property. This property says that if three pairs of corresponding sides are congruent, then we can prove the triangles overall are congruent.
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Problem 4
Short Answers:
h) Vertical Angles are congruent
i) ASA Congruence Property
j) CPCTC
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Explanations:
h) If two lines cross each other, then the angles opposite each other are congruent. To be congruent means they have the same angle measure.
i) ASA stands for Angle Side Angle. The first "Angle" refers to angle TRG = angle NAG. The "side" refers to the fact that RG = GA. The last "angle" refers to angle TGR = angle NGA
j) CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". A good analogy of this is that if we have two identical houses, then the front doors must be the same. The "house" is the "triangle"; the "front door" is analogous to the angles since its a part of the overall whole structure.
