Hypotenuse angle (HA) theorem (Proof & Examples)

Geometry might seem choose no laughing matter, however this lesson has an ext than one HA moment. That"s due to the fact that this is all about the Hypotenuse edge Theorem, or HA Theorem, which allows you come prove congruence of two best triangles using just their hypotenuses and also acute angles.

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What are best Triangles?

Before we start roaring with all the laughs this lesson bring (!), let"s make sure we have actually a firm knowledge of right triangles. A appropriate triangle is dubbed that since it has actually one appropriate angle (90°), which way the various other two angles must be acute (less than 90°).


This best angle limits the feasible measurements that the other two angles. With each other they must include up to 90° since all inner angles of any triangle -- right, scalene, obtuse -- must add to 180°. Individually the one best angle native 180° leaves just 90° come be common by the two continuing to be angles, make both of lock acute angles.

Can you watch what would take place if us knew something around one the those two acute angles? we would know something around the remaining angle. Let"s take a watch at exactly how that dram a function in the HA Theorem.

HA Theorem

Now that we have actually right triangles right in ours heads, let"s look in ~ the HA Theorem.


Congruence does no mean just somewhat alike; it way the 2 triangles will certainly be identical; every side and also every angle, equal in between the triangles. That"s a tall order, and we room claiming to get it simply by understanding one next (the hypotenuse) and one angle.

Remember, though, the we currently know a 2nd angle. We know the ideal angle that develops a square corner.

Here space two appropriate triangles, △ZAP and △HOT.


They are both facing with your hypotenuses come the right, which way their best angles space to the left -- HA! (A small touch the triangle humor.)

Proving the HA Theorem

Notice ∠A and also ∠O are appropriate angles, shown by the tiny square □ tucked right into the inner angles. We room told the the hypotenuses, ZP and also HT, room congruent, i m sorry is why they have actually the tiny matching hash marks. We are additionally told the acute ∠Z and ∠H are congruent, presented by their own hash marks.

If us knew just that lot geometry, we would certainly be stumped. We can say the six parts (three sides and also three angles) have actually only three parts congruent, and they space not all touching.

Look carefully -- ∠A and ∠Z space consecutive angle in ours left best triangle … uh … our best triangle on the left (HA feeling again). Those two angles execute not incorporate a known side between them. We have actually no idea if ZA is congruent to HO.


Check the end the continuing to be angles. ∠P and ∠T. What perform we know about them? We recognize they are congruent. Why?

They have to be congruent since of what we stated earlier. Given 2 of the angles, the 3rd angle is uncovered by subtracting the two provided angles from 180°. We execute not even need numbers because that ∠Z and also ∠H; they space congruent, for this reason ∠P and also ∠T are congruent.

So What?

So what, you say? If we recognize that all 3 angles room congruent, and we know that included sides between angles space congruent, then we have actually the ASA Postulate! Recall the ASA speak us:

Triangles room congruent if any type of two angles and their consisted of side are equal in the triangles.

Building off that handy appropriate angle, we settled two included angles, ~ above either side of the hypotenuse. Now we have all these congruences:

∠A ≅ ∠O (two best angles, i beg your pardon we used to deduce ∠P ≅ ∠T)∠Z ≅ ∠H (a given)Hypotenuse ZP ≅ hypotenuse HT (a given)∠P ≅ ∠T (deduced indigenous ∠Z ≅ ∠H and also ∠A ≅ ∠O)

The last three congruences are the ASA Postulate in ~ work. HA! us did some amazing detective work there.

All That?

Do you need to go with all the every time you want to display two right triangles space congruent? No. You have the right to use the HA Theorem! HA! (We told girlfriend this would have an ext than one HA moment.)

Instead that going with the an extensive process of detect the third angle congruent, hauling the end the ASA Postulate, and declaring the two right triangles congruent, you can easily apply the HA Theorem.

HA Theorem exercise Proof

You cannot show off the HA Theorem v something as an easy as two twin ideal triangles, charming together △ZAP and △HOT are. What around something trickier, like two right triangles seeming to slide previous each other, favor these:


These two ideal triangles were constructed from heat OA, intersected by heat FB, crossing at Point G.

See more: (-X)^2 Equals - Squares And Square Roots In Algebra

Right △FOG share a vertex, Point G, through △BAG. We view that ∠O and ∠A are ideal angles, and also the tiny hash point out tell united state hypotenuses FG and BG space congruent. What else space we told? Nothing!

Are you prepared to have actually a HA moment? We know sides OG and AG kind a right line, due to the fact that they are segments of line OA. We recognize that both appropriate triangles re-superstructure Point G, developing two internal angles (∠FGO and also ∠BGA). Those inner angles are vertical angles of 2 crossing lines! HA! upright angles space congruent. Now we have actually another collection of congruences. Let"s do a list:

∠FGO ≅ ∠BGAHypotenuse FG ≅ Hypotenuse GB

With simply the hypotenuse and one acute angle, we now release the strength of the HA Theorem and state that:


Lesson Summary

Though it might not have been a barrel of laughs, by trying out the HA Theorem friend are now able come recall and also state the Hypotenuse angle (HA) Theorem, show the HA Theorem"s connection to the Angle next Angle Theorem, and mathematically prove the HA Theorem.

Next Lesson:

HL Theorem

What you"ll learn:

After researching these instructions, illustrations and the video, you will certainly be able to:

Recall and also state the Hypotenuse angle (HA) Theorem show the HA Theorem"s connection to the Angle side Angle TheoremMathematically prove the HA Theorem