Review Sheet, and p. 239: 7-16; p. 281: 1, 2, 7, 10, 11; p. 827: 9-11; p. 855: 7, 10, 11
Review Sheet Hints:
1.) You can’t assume that <1, <2 and <3 add to 180. We want to show this is true using other known facts. Use the hint on the sheet, as well as the fact that a straight line has an angle measure of 180. So what can you say about <5, <2, and <4? From there, use substitution.
2.) The 3 angles of a triangle add up to 180. Create an equation, solve for x.
3.) Same hint as above. Be careful combining like terms though, x should be an integer, so if you get decimals you made a mistake.
4.) Draw a 5 sided shape. We use the triangle sum theorem (the three angles of a triangle add up to 180), by splitting up the 5 sided shape into 3 triangles, and then adding together the angles in each triangle to find the angles of the whole shape.
5.) Use what you found in number 4. What do the 5 angles of a pentagon add up to?
6.) Exterior angles always add up to the same number, 360. Once you find the missing exterior angle, you can use that to find x, since they are a linear pair.
7.) Maybe you remember the quick pattern if not. Find the third angle of the triangle, and use what you know about linear pairs (they add to 180) to find the exterior angle y. Maybe you will see the pattern after you do this.
8.) Same concept as 7. Find missing information, create an equation.
9.) Third angles theorem. What is given about the two triangles? What can you conclude. This conclusion allows you to create an equation, find x, and plug it back in.
10.) Just because they don’t start with the same structure (SSS, ASA, AAS, SAS) doesn’t mean they definitely aren’t congruent. Use the given info to find other information. You know two angles of a triangle. Any time you have that, go ahead and find the missing angle. Then what can you say? You can use ASA or AAS to show congruence in this case.
11.) Isolate lines BC and AD in a separate diagram. You could technically think of AB as a transversal, but that would give you Same Side Interior angles, which would be supplementary by the Same Side Interior Angles Theorem. You want to identify a transversal that gives you an angle pair that are congruent. Think about line segment AC. What kind of angle pair does that give you?
12.) a.) Note: give or use info about supplementary angle pairs. Think original and converse versions.
b.) Postulates: once again think about the original and converse version.
13.) Acknowledge your given info, and that is going to need to be combined with info that you can get from the linear pair theorem. Look at the diagram. Write statements that are true from the linear pair theorem. Use substitution (a few times) to create new equations.
14.) Since <DAC is congruent to <DCA, we know that DA is congruent to DC (prop. of Isosceles Triangles). From there, use definition of isosceles triangles for the other given info, and reflexive prop. This will be enough to show that two of the smaller triangles are congruent, and then you can use CPCTC and definition of bisect to show what we want to prove.
15.) Since you know angle X, you also know angle Z, and you can use those to find angle Y. (Similar to number 2)
16.) Focus on one triangle at a time. One you have 90 and 60 degree angles. What is the third angle? The other triangle is isosceles because of the tick marks on the sides, so you know the two missing angles are equal, and the other given angle is 80. x+x+80=180. After all of this, you use the angle addition postulate to find all of angle ABC.
17.) What is true when a segment bisects another segment? It splits it into two congruent halves. Mark that! Perpendicular tells you all of the angles at that intersection are 90 degrees. Two angles that have the same measure are congruent (def. of congruence). You also need to use the reflexive property to show that a side is congruent to itself. Then identify what triangle structure tells you that the given triangles are congruent.