Identify the congruence theorem of the following right triangles.
(LL, LA, HL, HA) ANSWER ASAP THIS IS 100 points
Answers
Answer:
Right triangles are aloof. They stand apart from other triangles, and they get an exclusive set of congruence postulates and theorems, like the Leg Acute Theorem and the Leg Leg Theorem. The Leg Acute Theorem seems to be missing "Angle," but "Leg Acute Angle Theorem" is just too many words.
Right Triangles
LA Theorem
LA Theorem Proof
LL Theorem
LL Theorem Proof
Examples
Identifying Property of Right Triangles
Right triangles get their name from one identifying property:
A right triangle contains one interior angle measuring 90°.
It must, of course, be a triangle, meaning it is a three-sided polygon. It cannot have two interior right angles because then it would not be a triangle.
Right triangles can be any size, so long as you get your required three sides and three interior angles, one of which must be 90°. The triangle can face any direction. "Right" does not refer to direction; it comes from the Latin angulus rectus or "upright angle."
Notice the elegance of the unspoken consequence of one right angle: the other two angles of a right triangle must each be acute, or less than 90° each. In fact, they will be complementary, meaning they will add to 90° (not free as in complimentary peanuts).
Right triangles have hypotenuses opposite their right angles. Hypotenuses are sides. The other two sides are called legs, just as an isosceles triangle has two legs.
Because all right triangles start with one right angle, when you try to prove congruence, you have less work to do. Mathematicians always enjoy doing less work.
Leg Acute (LA) Theorem
The Leg Acute Theorem, or LA Theorem, cannot take its proud place alongside the Los Angeles Rams, Los Angeles Angels, or Anaheim Ducks (wait, what?). The LA Theorem has little to do with The City of Angels.
The LA Theorem states: If the leg and an acute angle of one right triangle are both congruent to the corresponding leg and acute angle of another right triangle, the two triangles are congruent.
If you recall our freebie right angle, you will immediately see how much time we have saved, because we just re-invented the Angle Side Angle Postulate, cut out an angle, and made it special for right triangles.
Proving the LA Theorem
Below are two run-of-the-mill right triangles. They look like they are twins, but are they? We have labeled them △WIT and △FUN and used hash marks to show that acute ∠W and acute ∠F are congruent.
We have also used hash marks (or ticks) to show sides IW ≅ UF. But, we have also used □ to identify their two right angles, ∠I and ∠U.
LA Theorem
Before you leap ahead to say, "Aha, The LA Theorem allows us to say the triangles are congruent," let's make sure we can really do that.
Right angles are congruent, since every right angle will measure 90°. Let's review what we have:
∠W ≅ ∠F (given)
IW ≅ UF (given)
∠I ≅ ∠U (right angles; deduced from the symbol □, right angle)
That, friend, is the Angle Side Angle Postulate of congruent triangles. To refresh your memory, the ASA Postulate says two triangles are congruent if they have corresponding congruent angles, corresponding included sides, and another pair of corresponding angles.
We think we know what you're thinking: what if we had two different sides congruent, like IT ≅ UN? What then?
Well, what of it? If you know ∠W ≅ ∠F are congruent, then you automatically know ∠T ≅ ∠N, because (and this is why right triangles are so cool) those two acute angles must add to 90°! If one pair of interior angles is congruent, the other pair has to be congruent, too! So you still have Angle Side Angeles -- er, Angle.
The theorem is called Leg Acute so you focus on acute legs, using those congruent right angles as freebies, giving you two congruent angles to get Angle Side Angle.
The Leg Leg (LL) Theorem
But, friend, suppose you have two right triangles that are not cooperating? You have two pairs of corresponding congruent legs. That's it. They refuse to cough up anything else. Are you going to use the Leg Acute Theorem? Of course not! Leave it in your geometer's toolbox and take out the sure-fire LL Theorem.
The Leg Leg Theorem says Greg Legg played two seasons with the Philadelphia Phillies -- nope; wrong Leg.
The Leg Leg Theorem says: If the legs of one right triangle are congruent to the legs of another right triangle, the two triangles are congruent.
How can this be?
Proving the LL Theorem
Here we have two right triangles, △BAT and △GLV.
LL Theorem
We have used ticks to show BA ≅ GL and AT ≅ LV. Do we know anything else about these two triangles?
Sure! We know that ∠A ≅ ∠L because of that innocent-looking little right-angle square, □, in their interior angles. It may look like first, second or third base, but it is better than that.
What do we have now?
BA ≅ GL (given)
∠A ≅ ∠L (from □)
The triangles have
1. LA congruence.
2. HL congruence.
3. LA congruence.
4. HA congruence.
5. HA congruence.
6. LA congruence.
7. LL congruence.
8. LL congruence.
9. HA congruence.
10. LL congruence.
11. HL congruence.
Given that,
In the picture we can see the triangles.
We have to find the congruence theorem of the right triangle (LL, LA, HL, HA).
We know that,
LL - If the legs of one right triangle are congruent to the corresponding legs of another right triangle then the triangles are congruent to each other with LL congruence.
LA - If the one leg and acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle then the triangles are congruent to each other with LA congruence.
HA - If the hypotheses and acute angle of one right triangle are congruent to the hypotheses and corresponding acute angle of another right triangle then the triangles are congruent to each other with HA congruence.
HL - If the hypotheses and legs of one right triangle are congruent to the hypotheses and corresponding legs of another right triangle then the triangles are congruent to each other with HL congruence.
Therefore,
1. LA congruence.
2. HL congruence.
3. LA congruence.
4. HA congruence.
5. HA congruence.
6. LA congruence.
7. LL congruence.
8. LL congruence.
9. HA congruence.
10. LL congruence.
11. HL congruence.
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