today's lesson is on triangle congruence by angle side angle and angle angle side take a minute to read over The Learning goal and the scale find where you are on the scale before we begin the lesson in our last lesson we learned we can prove two triangles are congruent without having to show all their corresponding parts are congruent we already know that triangles are congruent if two pairs of sides and the included angles are congruent that's our side angle side postulate we can also prove triangles congruent using other groupings of angles and sides we can prove two triangles congruent by using one pair of corresponding sides and two pairs of corresponding angles let's have a look at the angle side angle postulate if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle then the two triangles are congruent in this triangle angle a is congruent to angle D of this triangle side AC is congruent to side DF and angle C is congruent to angle F here we have angle side angle angle side angle therefore triangle ABC is congruent to Triangle DF by the angle side angle postulate in example one we will use angle side angle which two triangles are congruent by angle side angle explain well we know that in order to use the angle side angle postulate congruent corresponding sides must be included between the two congruent corresponding angles so let's have a look at triangle Su UV here we have angle U side U UV and angle V so side UV is included between the two given angles in Triangle o n e we have angle e we have side EO and we have angle o side EO is included between the two angles finally in Triangle TW wa we have two angles and side w a is not included between angles T and W therefore this is not angle side angle the two triangles that we can prove congruent by angle side angle are triangle SUV and triangle NE EO notice that we listed the names of the triangles in corresponding congruent order pause the video and do UT try number one which two triangles are congruent by angle side angle explain when using angle side angle we want the congruent corresponding sides to be included between the two angles so here in Triangle h o g we have side HG included between angle H and angle G in Triangle f i n we have side FN but it is not included between angles n and I however in Triangle c a t we have included side CA between angle C and angle a therefore triangle h o g is congruent to Triangle a t c by angle side angle and notice that we listed our corresponding part in order in example two we will write a proof using angle side angle the members of a teen organization are building a miniature golf course at your Town's Youth Center the design plan calls for the first hole to have two congruent triangular bumpers prove that the bumpers on the first hole shown at the right meet the conditions of the plans we want to prove that triangle ABC is congruent to Triangle DF let's start with the given information that side AB is congruent to side de and that angle a is congruent to angle D next we can see that angle B and angle e are right angles that is also given information in step three we know that angle B is congruent to angle e because all right angles are congruent in step four since we have an angle a side and an angle congruent to an angle a side and an angle of both triangles we know that triangle ABC is congruent to Triangle DF by the angle side angle postulate pause the video and do UT try number two here we want to prove that triangle AB C is congruent to Triangle AED D let's start with our given information that angle C A is congruent to angle D AE and side ba is congruent to side EA next let's look at angle B and angle e that was given that they are both right angles for statement three angle B is congruent to angle e because all right angles are congruent and for statement four we can go ahead and say that triangle ABC is congruent to Triangle a d by angle side angle angle side angle we can also prove triangles congruent by using two angles and a non-included side let's have a look at the angle angle side theorem if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle then the triangles are congruent here angle a is congruent to angle D angle B is congruent to angle e and side AC is congruent to side DF notice that side a is not included between angles A and B side DF is not not included between angles D and angle e so triangle ABC is congruent to Triangle DF have a look below at two other examples of the angle angle side theorem let's take a minute to prove the angle angle side theorem we want to prove triangle ABC is congruent to Triangle DF notice that the given information is that angle a is congruent to angle D angle B is congruent to angle e and non-included sides AC C and DF are congruent notice that side AC is corresponding to side DF because they are between the angles with one Arc and no arcs since the given information is angle angle side angle angle side let's see if we can figure out another way to prove these triangles congruent therefore proving that angle angle side theorem does work let's start with statements 1 and two being our given information angle a is congruent to angle D and angle B is congruent to angle e because we know two angles of one triangle are congruent to two angles of another by the third angles theorem we can state that angle C is congruent to angle f for statement 4 let's use the given information that side AC is congruent to side DF now we have angle side angle congruent to an angle side and an angle of both triangles therefore triangle ABC is congruent to Triangle DF by angle side angle postulate since we started off with the given information of two angles and a non-included side of one triangle being congruent to two angles and a non-included side of another and prove the triangles congruent by angle side angle we know that the angle angle side theorem also works in example three we will write a proof using the angle angle side theorem let's start with the given information that angle m is congruent to angle K and that side w n is parallel to side RK since we have sides that are parallel let's look for alternate interior angles here we can see that angle MWR is congruent to angle KRW by the alternate interior angles theorem we also know that side w r is congruent to side r w by the reflexive property of congruent ruent because we have an angle an angle and a non-included side congruent to an angle an angle and a non-included side triangle wmr is congruent to Triangle rkw by the angle angle side theorem let's see if we can prove these two same triangles congruent by using angle side angle instead let's let start with the same pieces of given information remember angle MWR is congruent to angle KRW because they are alternate interior angles now by the third angle theorem we know that angle Mr W is congruent to angle kwr side W is congruent to side RW because of the reflexive property of congruent now that we have two angles and an included side congruent to two corresponding angles and an included side we know that triangle wmr is congruent to Triangle rkw by angle side angle pause the video and do UT Trine number three here we want to prove that triangle SRP is congruent rent to Triangle qrp let's start with the given information that angle s is congruent to angle q and that side RP bisects angle SRQ because we know the definition of BCT and side RP bisects angle SRQ we know that angle SRP is congruent ruent to angle qrp we also know that by the reflexive property of congruence side RP is congruent to side RP since we have two angles and a non-included side congruent to two angles and a non-included side triangle SRP is congruent to Triangle qrp by the angle angle side theorem in example four we will determine whether triangles are congruent here we want to use the diagram at the right which of the following statements best represents the answers and justification to the question is triangle B congruent to Triangle u in both triangles we have two angles and a non-included side two angles and a non-included side that are congruent however side BF is not corresponding to side OT side BF is corresponding to side uo therefore we cannot prove that the triangles are congruent because side BF is not corresponding to side to the answer is B pause the video and do UT try number four here we have two corresponding sides and two corresponding angles of two triangles that are congruent we know that angle p r a is congruent to angle SRI I because they are vertical angles now since we have an angle another angle and a non-included side congruent to an angle another angle and a non-included side that these two triangles are congruent by angle angle side now is your chance to see how well you've understand the lesson pause the video and do the lesson check don't forget to check your answers on the next slide if there are any questions you are unsure of please ask me tomorrow in class if you rock the lesson check take a shot at the challenge I'm sure you can get it done take a minute to reread the learning goal in the scale see if you've climbed the scale any higher since going over the lesson