today's lesson is on congruence in right triangles take a minute to read over their learning goal in scale find where you are in the scale before we start the lesson in this diagram two sides and a non-included angle of one triangle are congruent to two sides and a non-included angle of the other triangle it is very obvious that the triangles are not congruent so we can conclude that the side side angle is not a valid method for proving two triangles congruent however this method does work in special cases of right triangles where the right angles are the non-included angles in a right triangle the side opposite the right angle is called the hypotenuse it is the longest side of the triangle the other two sides are the legs remember we can prove that two triangles are congruent without having to show that all of their corresponding parts are congruent today we're going to prove right triangles congruent by using one pair of right angles a pair of hypotenuses and a pair of legs the hypotenuse leg theorem says that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle then the triangles are congruent are you ready to prove the hypotenuse leg theorem to prove the hypotenuse leg theorem we will need to draw an auxiliary line to make a third triangle let's start by redrawing triangle XYZ and on triangle XYZ draw Ray zy now let's mark Point s so that the length of segment Ys is congruent to the length of segment QR connect points X and S so that triangle X Ys is congruent to Triangle pqr since corresponding parts of congruent triangles are congruent segment p r is congruent to segment XS by the transitive property of congruence segment XS is congruent to segment XZ by the isoceles triangle theorem angle s is congruent to angle Z so triangle x y s is congruent to Triangle XYZ by the angle angle side theorem to use the hypotenuse leg theorem the triangles must meet three condition conditions first there must be two right triangles second the triangles have to have congruent hypotenuses and third there's one pair of congruent legs in example one we will use the hypotenuse leg theorem on the basketball backboard bracket shown below angle a d c and angle BDC are right angles and side AC is congruent to side BC are triangles a DC and triangle BDC congruent since both of the triangles are right triangles and we have two congruent hypotenuses now we just need one set of congruent legs by the reflexive property of congruence side DC is congruent to side DC that is our congruent pair of legs so by the hypotenuse leg theorem triangle ADC is congruent to Triangle BDC pause the video and do UT try number one in part A we were to prove that triangle PS is congruent to Triangle rpq since angles P RS and rpq are both right angles triangle P RS and triangle rpq are right triangles it is also given that side SP is congruent to side QR since these sides are opposite the right angles both of the hypotenuses are congruent by the reflexive property of congruence side PR R is congruent to side RP since we have two right triangles with congruent hypotenuses and one pair of congruent legs triangle PS is congruent to Triangle rpq by by the hypotenuse leg theorem in Part B your friend says suppose you have two right triangles with congruent hypotenuses and one pair of congruent legs it does not matter which leg in the first triangle is congruent to which leg in the second triangle is your friend correct explain yes your friend is correct to satisfy the hypotenuse leg theorem you just have to meet three criteria first is that you have two right triangles second is that the hypotenuses are congruent and third that you have one pair of congruent legs since this scenario meets all three criteria your friend is correct and the triangles will be congruent in example two we will write a proof using the hypotenuse leg theorem we want to prove that triangle ABC is congruent to Triangle D let's start with the given fact that segment be by BCT segment a d at Point C by the definition of bisect we know that segment AC is congruent to segment DC now let's use the second and third given statements that segment AB is perpendicular to segment BC and segment D is perpendicular to segment EC by the definition of perpendicular angle B and angle e are right angles since all right angles are congruent angle B is congruent to angle e now let's use the final given statement that segment AB is congruent to segment D we can now say that triangle ABC is congruent to Triangle DEC by the hypotenuse leg theorem we have two right triangles with congruent hypotenuses and one pair of congruent legs pause the video and do UT try number two here we want to prove that triangle CBD is congruent to Triangle e they tell us that side CD is congruent to side EA so that is given we are also given that segment a d is the perpendicular bis sector of segment C by the definition of perpendicular we know that angle CBD and angle EB are right angles because all right angles are congruent angle CBD and angle EB are congruent by the definition of bis sector we know that segment CB is congruent to segment EB and finally we can say that triangle CBD is congruent to Triangle ebaa by the hypotenuse leg theorem we have two right triangles with congruent hypotenuses and one pair of congruent legs now is your chance to see how well you understand the lesson pause the video and do the lesson check don't forget to check your answers on the next slides go ahead and check the answers for questions 1 through 4 now check your answers for numbers 5 through 7 if you've missed any and you're not sure why please be sure to ask me in class if you rock the lesson check go ahead and try the challenge I'm sure you can do this one too now take another minute to reread the learning goal in the scale have you climbed any higher on the scale since we covered the lesson