1. What is the sum of the measures of the angles in a triangle? Write the proof (Hint: it involves creating a parallel line.)
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1 riangle asics irst: Some basics you should already know. eometry What is the sum of the measures of the angles in a triangle? Write the proof (Hint: it involves creating a parallel line.) 2. In an isosceles triangle, the base angles will always be. he proof of this generally involves some information we will review today, but here it is two ways: riangle is congruent to riangle (side-angle-side) therefore angle = angle. Not satisfied? dd some lines: = riangle = riangle (SS) herefore triangle = (subtraction) Which makes angle = 3. If exactly two angles in a triangle are equal then it must be. (this is the converse of #2) 4. What is the relationship between an exterior angle of a triangle and the sum of the remote interior angles? Prove with just a sentence or two. remote interior angles adjacent interior angle exterior angle 5. In triangle XYZ: XY=6inches, YZ=9 nches, and XZ=11 inches. Which is the largest angle: X, Y, or Z? he smallest? 6. Which of the following sets of numbers could NO represent the three sides of a triangle? How many scalene triangles have sides of integral (integer) length and perimeter less than 15?
2 riangle ongruencies Name Period eometry 4.4 You have probably already heard of most of the triangle congruence shortcuts. oday we will construct several triangles to demonstrate the shortcuts we can use to show two triangles are congruent. H igures are considered congruent if they are exactly the same. If you can slide, rotate, or reflect one figure so that it is exactly the same as another, the two figures are considered congruent. 1. SSS: Side-Side-Side Use the three sides above to construct a triangle (begin with H). ompare it to the ones your classmates drew. oes SSS demonstrate congruence? 2. SS: Side-ngle-Side Use, angle and H to construct a triangle. ompare it to the triangle your classmates drew. oes SS demonstrate congruence? 3. S: ngle-side-ngle Use angle, segment H, and angle H to construct a triangle. ompare it to the triangle your classmates drew. (Is S a congruence shortcut? Why or why not?) 3+. S: Side-ngle-ngle H 4. SS: Side-Side-ngle Use angle, segment H, and segment H to construct a triangle. ompare it to the triangle your classmates drew. oes SS demonstrate congruence? Is it possible to draw more than one triangle using angle, segment H, and segment H? H
3 riangle ongruencies HL and LL congruence: Use the following segments again. Name Period eometry 4.5 H 1. LL: Leg-Leg (or right triangles.) onstruct Right angle H. onnect H. ompare your triangle to the ones your classmates drew. Which congruence shortcut is this identical to? 2. HL: Hypotenuse-Leg (or right triangles.) onstruct right angle H on segment H. Use length to complete right triangle H. ompare your triangle to the ones your classmates drew. Is this similar to any of the congruence shortcuts on the opposite side of this page?
4 Using ongruence Shortcuts eometry 4.5 etermine which of the following pairs of triangles are congruent and why: riangles are not necessarily to scale. = by. =? o 55 o 55 o =? 10 =? 55 o 80 o 35 o 60 o 35 o 60 o 80 o =? = =? 80 o 80 o
5 Using ongruence Shortcuts eometry 4.5 etermine which of the following pairs of triangles are congruent and why: riangles are not necessarily to scale. = by. =? =? =? 4cm o o 5cm 89 o =? is the centroid. = =? =?
6 Name Period Proof/ eginning Proof Writing eometry 4.5 Mathematical Proof takes an accepted set of facts and properties to demonstrate something to be true. In a two-column proof, statements are made on the left and justifications are made on the right. x. iven: and = Prove: = 1. = 1. iven 2. = 2. iven iven 4. = 4. I 5. = 5. I 6. = 6. ransitive Property of ongruence 7. is isosceles. 7. ase ngles are ongruent (onverse of Isos. heorem) 8. = 8. efinition of Isosceles 9. = 9. SS (1,2,8) Some common justifications you will be using in your proofs: lternate Interior ngles (I) orresponding ngles () efinition of. (midpoint, square, kite, vertical angles, bisector, etc.) SSS, S, SS, S, HL, LL Same Segment or Same ngle (ex. If you said =. his will later be called the Reflexive Property of ongruence, but that is not necessary now.) Vertical ngles Linear Pair etc. On the back, record any new justifications that we learn so that you can have a list to use when writing proofs.
7 P eometry 4.6 If you can show that two triangles are congruent, then their corresponding parts are also congruent. P: orresponding Parts of ongruent riangles are ongruent We will use this shortcut when writing wo-olumn Proofs. iven: is the midpoint of segment. and are parallel. ngle and ngle are congruent. Prove: =. In a two column proof, statements are made in the left column, and justifications for those statements are on the right. 1. egin with the given information. 2. Work through the diagram to determine whether the conclusion can be reached. 3. Organize the steps carefully as in the example below, including the given information. X. iven: is the midpoint of segment. and are parallel. ngle and ngle are congruent. Prove: =. 1. is the midpoint of. 1. iven 2. = 2. efinition of midpoint 3. and are parallel. 3. iven 4. = 4. orresponding angles 5. = 5. iven 6. = 6. S congruence (# 2, 4, 5) 7. = 7. P
8 P Write a two-column proof for each: iven: = Prove: = iven: = bisects Prove: = 90 o eometry 4.6 or each problem below, some of the given information is included in the diagram. iven: = bisects Prove: = iven: Prove: = iven: is the midpoint of is the midpoint of = Prove: =
9 lowchart Proofs eometry 4.7 lowcharts can be used to explain the logical organization of a proof. In a flowchart proof, statements are placed within boxes, with the justification below the box. rrows connect statements. he arrow can be read as the word therefore. iven: = = Prove: = = iven = ongruent sides of an isosceles = Vertical angles = S = iven iven: = Prove: = his can be done many ways, try to find the easiest.
10 Name Period Practice Proofs Half-Quiz eometry 4.7 omplete the following Proofs by filling in the missing blanks: R iven: R = R Prove: = R 1. R lternate Interior ngles = S ongruence (3, 2, 4) P S R iven: S = S S = Prove: S = 1. S = S P iven Same ngle 5. S = P
11 Name Period Practice Proofs Half-Quiz eometry 4.7 omplete the following Proof by filling in the missing blanks: L iven: M = R M R Prove: M = R M = R P M = R MR = RM M = R P SS R = M
12 Special Proofs eometry 4.8 One shortcut: or several the following proofs, we will shorten some steps by using the following theorem: If two angles are both linear and congruent, then they are right angles. In our proofs, the justification will look like: 1. XYZ = 90 o 1. ongruent Linear ngle (with WYZ). Proofs involving special triangles. Use a two-column or flowchart proof for each: 1. Prove that the bisector of the vertex angle in an isosceles triangle is also the median. 2. Prove that the altitude from the vertex of an isosceles triangles is also an angle bisector. 3. In a given circle, prove that if a radius bisects a chord then the chord and radius are perpendicular. 4. xplain (too long for a formal proof) why the incenter, circumcenter, orthocenter, and centroid are all the same point in an equilateral triangle. Proofs involving quadrilaterals. Use a two-column or flowchart proof for each: 1. Prove that the diagonals in a square are angle bisectors. 2. Prove that the diagonals in a parallelogram are of equal length. 3. xplain how you could prove that the diagonals in a parallelogram bisect each other. 4. Prove that the diagonals in a rhombus are perpendicular (to each other). onfused about one of these?
13 P and Proofs Prove each of the following using P. Write a two-column proof for: Name Period iven: = Prove: = eometry Re omplete a flowchart proof for the following: iven: = Prove: =
14 ongruence Review Name Period eometry Re etermine which of the following pairs of triangles are congruent and why: riangles are not necessarily to scale. Write cannot be determined where appropriate. = by = by = by = by 4cm 61 o 30o 89 o 5cm M = by P = by M P R N R = U by R PI = P O by I S
15 Name Period P and Proofs eometry Re omplete a proof for each: Use a separate sheet if needed. iven: = = = Prove: = S iven: R R = SR N Prove: SN = N hallenge: riangles and are equilateral. Prove that =.
16 ongruence Review eometry etermine which of the following pairs of triangles are congruent and why: riangles are not necessarily to scale. Write cannot be determined where appropriate. M = by P = by M N Name Period P R Re O = O by PI = O by I P = by? = by? 55 o U = by 55 o R U Y
17 Proofs Practice Name Period eometry 4.8 or each of the following: Sketch the situation and label all given information. reate a two-column proof for the given statement. 1. Prove that in a given circle, if chords and are congruent, then angles and are also congruent. 2. Segments and bisect each other. Prove that segments and are parallel. Hints or ack: 1. You will need to use the base angles of an isosceles triangle. 2. You will use vertical angles. 3. You will prove two triangles that look congruent are congruent. 4. You will use ongruent Linear ngles.
18 Name Period Proofs Practice eometry 4.8 hallenge: In the figure below, = and bisects angle. Prove Hints can be found on the front.
19 est Review: eometry 4.8 ill-in the blanks for each triangle congruence below. Write cannot be determined where appropriate. = by? H H = H H = = H = H = = by? by? by? by? by? List the sides below in order from shortest to longest. (not to scale) 35 o 10 o b e a 90 o g 45 o 125 o 55 o 69o 55o o f 125 d 56 c o 40 o 15 o
20 Name Period Proofs Practice est eometry 4.8 ill-in the blanks for each: Write annot be determined where appropriate. OY = by P = by N Y P O R L O M PIN = by PI = by I N P P K R I Y = by M M = by N U M = by = by
21 Name Period Proofs Practice est eometry 4.8 omplete the flowchart proof below: iven: = = = H Prove: H = Write a two-column proof to prove that if congruent segments and are parallel, then =. Include a small sketch and all givens. Use as few spaces as possible. note: <
22 est Review eometry 4.8 on t uess! he diagrams below are misleading and force you to ONLY US WH YOU KNOW. Just because two triangles look congruent does not mean that they can be proven congruent using S, SSS, SS, S, etc. Practice: Which pair(s) of triangles is congruent? L H J K M P Practice: Which pair(s) of triangles is congruent? I N J L M N P H K S R Practice: Prove: is the Prove: JH = H midpoint of. J K H
23 est Review Practice: Misleading diagrams. Which pairs of triangles are congruent N why? Name Period eometry K L H J M P Q U X Y N R S V W Z Practice: Misleading iagrams. Which pairs of triangles are congruent N why? H J V U X K W Y Practice: On back, write a 2-column proof to show that the diagonals of a kite are perpendicular to each other. K X I
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