Download Notes on Midsegments and ALL Triangle

NOTES
Midsegment of a Triangle: a segment whose endpoints are the midpoints of two sides.
ACTIVITY 1: TRIANGLE MIDSEGMENTS
A
1. Draw Δ ABC.
2
C
1
4
3
B
2. Find the midpoints, M and N, of sides AB and AC. Then draw MN, the midsegment.
3. Measure and record MN and BC on your paper. What is the relationship between their
lengths?
4. Measure Ð1 and Ð2. Measure Ð3 and Ð4. What do your measurements suggest about
BC and MN? What postulate or theorem allows you to draw this conclusion?
5. Rewrite and complete the following conjecture:
Triangle Midsegment Conjecture
A midsegment of a triangle is _______________ to a side of the triangle and has a measure equal
to ___________________ of that side.
WHAT DOES IT MEAN FOR TWO FIGURES
TO BE CONGRUENT?
They must have the same:
• SIZE
• SHAPE
HERE’S THE SITUATION…
Prior to the start of a sailboat race, you (the judging official) must certify that all of the sails are the
same size. Without unrigging the triangular sails from their masts, how can the official (you)
determine if the sails on each of the boats are the same size?
•
With your group discuss and write down how you would go about doing this?
•
Over the next couple of classes we will be learning some geometry tricks (concepts) involving
triangles that will help us answer the above question.
•
Hand out materials
ACTIVITY 2: SSS POSTULATE
•
Using these three objects, create a triangle. (The three sides being the ruler, unsharpened
pencil and straightedge of the protractor.)
•
Compare your triangle with your group members triangles.
•
What do you notice?
•
Did everyone create the same triangle?
•
Are all of your triangles congruent?
• Yes
•
Why?
• All of the parts are the same or congruent.
•
Notice that we did not even pay any attention to the angles and they “took care of themselves”
•
Create another triangle using the three objects, but this time only using 8 inches of the ruler for
one of the sides.
•
Are all of your triangles congruent again?
• Yes
•
With your group discuss how we can use this concept to relate back to our initial problem with
the sailboats.
SSS (SIDE-SIDE-SIDE) POSTULATE
If the sides of one triangle are congruent to the sides of
another triangle, then the two triangles are congruent.
ACTIVITY 3: SAS POSTULATE
1. Draw a 6 cm segment.
2. Label it GH.
3. Using your protractor, make  G = 60.
4. From vertex G, draw GI measuring 7 cm long.
5. Label the end point I.
6. From the given information, how many different triangles can be formed?
7. Form  GHI.
8. Is your  GHI congruent to your group members  GHI.
9. What information was used to create this triangle?
10. Draw another segment this time 10 cm long.
11. Label it XY.
12. Using your protractor, make  X = 45.
13. From vertex X, draw XZ measuring 5 cm long.
14. Label the end point Z.
15. How many different triangles can be formed?
16. Form  XYZ.
17. Is your  XYZ congruent to your group members  XYZ?
18. What information was used to create this triangle?
SAS (SIDE-ANGLE-SIDE) POSTULATE
If two sides and the included angle in one triangle are
congruent to two sides and the included angle in another
triangle, then the two triangles are congruent.
ASA (ANGLE-SIDE-ANGLE) POSTULATE
If two angles and the included side in one triangle are
congruent to two angles and the included side in another
triangle, then the two triangles are congruent.
PRACTICE
In each pair below, the triangles are congruent. Tell which triangle congruence
postulate allows you to conclude that they are congruent, based on the markings
in the figures.
AAS (ANGLE-ANGLE-SIDE) POSTULATE
If two angles and a nonincluded side of one triangle are
congruent to the corresponding angles and nonincluded
side of another triangle, then the triangles are
congruent.
PRACTICE
Which pairs of triangles below can be proven to be congruent by the AAS
Congruence Theorem?
TWO OTHER POSSIBILITIES
• AAA combination—three angles
• Does it work?
• SSA combination—two sides and an angle that is not between them (that is,
an angle opposite one of the two sides.)
SPECIAL CASE OF SSA
When you try to draw a triangle for an SSA combination, the side opposite the
given angle can sometimes pivot like a swinging door between two possible
positions. This “swinging door” effect shows that two triangles are possible for
certain SSA information.
A SPECIAL CASE OF SSA
If the given angle is a right angle, SSA can be used to prove congruence. In
this case, it is called the Hypotenuse-Leg Congruence Theorem.
REVIEW OF RIGHT TRIANGLES
HL (HYPOTENUSE-LEG) CONGRUENCE THEOREM
If the hypotenuse and a leg of a right triangle are
congruent to the Hypotenuse and a leg of another right
triangle, then the two triangles are congruent.
OTHER RIGHT TRIANGLE THEOREMS
LL (LEG-LEG) Congruence Theorem
If the two legs of a right triangle are congruent to the corresponding two legs of another right
triangle, then the triangles are congruent.
Right Triangle version of ______
LA (LEG-ANGLE) Congruence Theorem
If a leg and an acute angle of a right triangle are congruent to the corresponding leg and
acute angle of another right triangle, then the triangles are congruent.
Right Triangle version of ______
OTHER RIGHT TRIANGLE THEOREMS
HA (HYPOTENUSE-ANGLE) Congruence Theorem
If the hypotenuse and an acute angle of a right triangle are congruent to the corresponding
hypotenuse and acute angle of another triangle, then the triangles are congruent.
Right Triangle version of ______
HL (HYPOTENUSE-LEG) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the corresponding hypotenuse
and leg of another right triangle, then the triangles are congruent.