Answer:
No
Explanation:
"Looking" at our triangles
If Angle D is congruent to Angle G, and angle E is congruent to Angle H, lets call those the first and second vertices of each triangle. If we're trying to prove the triangles congruent, we're trying to prove Triangle DEF is congruent to Triangle GHJ (in that order, since we're already given that the first and second angles correspond between triangles).
Could these triangle be congruent? Yes! As a quick example, imagine two Equilateral triangles with side lengths 5. They are congruent, and they do have the first angle and second angle corresponding pairs congruent, and one of those sides from triangle 1 does match the length of one of the sides in triangle 2. ...and they are congruent triangles.
The problem with the given scenario is that the side they give for triangle 1 (between vertex 1 and 2) is not the corresponding side of triangle 2 (between vertex 2 and 3). This makes it so that we cannot guarantee that the triangles are congruent.
As can be seen in the attached image, I've made Side DE length 2, and put it on a coordinate plane so it's easy to see. Side HJ is also definitely length 2. Clearly, those triangles are not congruent.
When would triangles be congruent?
Going back to the definition of congruent triangles, three corresponding angles and three corresponding sides must be congruent. That's 6 pairs of parts needed to be proven congruent to prove triangles congruent!
Fortunately, there are a number of theorems that can be proven to lessen the amount of pairs of parts needed to be proven congruent and still guarantee that the resulting triangles must be congruent.
There are exactly 5 cases:
It looks like a bunch of alphabet soup, but notice that all of them require 3 pieces of information (even HL, which is Hypotenuse-Leg... since there's a Hypotenuse, it must be a right triangle, so there are a pair of corresponding congruent right angles that are congruent that they don't talk about)
In order to remember which items work, remember that you MUST know 3 parts, and think about all of the way you could know 3 parts:
Scenario 1: Have all 3 side pairs (SSS)
There is only one way that all three sides can be equal, and yes, that is one of the triangle congruence shortcuts.
Scenario 2: Have 2 side pairs & 1 angle pair
There are two option here.
- two sides and the angle between them (SAS), or
- two sides and an angle NOT between them (...nope).
Two sides with the angle between them is SAS (note the angle is between them).
Two sides with the angle not between them is ... (well, that can get you in trouble). Conveniently, that is NOT one of the triangle congruence theorems. So, stay out of trouble, and don't use this for triangle congruence.
Interestingly, HL, is a special case of this "trouble case". Note that if you have the Hypotenuse and a leg, that's two Sides. The right Angle is NOT the angle between those two sides (because the hypotenuse is always across from the right angle). When that angle is 90 degrees, then the set of two sides and the angle not between them DOES work as sufficient to prove triangle congruence, and it's the only time that it's sufficient.
Scenario 3: Have 1 Side and 2 Angles
Again, there are two option here:
- two Angles and the Side between them (ASA), or
- two Angles and a Side NOT between them (AAS).
In either scenario, this is sufficient to prove the triangles are congruent.
Scenario 4: Have all 3 angles (AAA)
This definitely won't be enough to prove congruence. Zoom in. It's got the same angles, but not the same lengths.
So, only 5 short ways to prove triangles are congruent, and all of them require 3 parts of one triangle corresponding to 3 parts of another, each of the 3 pairs must be congruent.