How to Prove Triangles Congruent – SSS, SAS, ASA, AAS Rules

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule. In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

 

 

Side-Side-Side (SSS) Rule

Side-Side-Side is a rule used to prove whether a given set of triangles are congruent. 

The SSS rule states that 

If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ.

 

 

Side-Angle-Side (SAS) Rule

Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent. 

The SAS rule states that 

If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is an angle formed by two given sides.

    Included Angle                         Non-included angle

For the two triangles below, if AC = PQ, BC = PR and angle C = angle P , then using the SAS rule, triangle ABC is congruent to triangle QRP

 

 

Angle-Side-Angle (ASA) Rule

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent. 

The ASA rule states that 

If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent.

Angle-Angle-Side (AAS) Rule

Angle-angle-side is a rule used to prove whether a given set of triangles are congruent. 

The AAS rule states that 

If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagrams below, if AC = QP, angle A = angle Q, and angle B = angle R, then triangle ABC is congruent to triangle QRP.

 

 

The following video will explain three ways to prove triangles congruent – A lesson on SAS, ASA and SSS,

 

Using Two Column Proofs to Prove Triangles Congruent

Triangle Congruence by SSS - How to Prove Triangles Congurent
Side Side Side Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Triangle Congruence by SAS – How to Prove Triangles Congurent
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Prove Triangle Congruence with ASA Postulate
Angle Side Angle Postulate
It 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.

Triangle Congruence by AAS Postulate
Angle Angle Side Postulate
It two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

 

 

 

 

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Tags: Aas Rules, Prove Triangles Congruent, Rules, Triangles Congruent

The Choice, a blog related to college admissions from the New York Times, has started providing posts to help students in India.

India Ink provides college admissions information specifically directed at some of the issues facing international students in general, and Indian students in particular.  The first few articles are from admissions officers at highly selective colleges in the US.

For my readers in India, this can be a good resource to provide general information about college admissions in the US. Of course, you can also contact me with your questions to get more specific answers.

 

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Edinburg CISD is hoping a 3,700-foot barrier will be the answer to keeping students at Harwell Middle School and Avila Elementary School safe from gunfire on nearby ranches.

School district officials and Hidalgo County commissioners are drawing up an interlocal agreement to spend anywhere from $140,000 to $280,000 dollars on the project.

Officials told Action 4 News that the county has agreed to provide the district with dirt, equipment, and fuel to get the ball rolling.

The agreement also states the county will provide labor for the spreading, compacting, shaping and  fine grading for approximately 3,400 feet of the barrier wall.

The district will be responsible for the construction of the wall and maintaining it.            

The wall will completely cover the north side of both schools and extend to the east and western portion of both campuses.

School district officials told Action 4 News that these changes are to better protect the students and they’re hoping it will ease the parents minds.

This agreement will not be made official until it goes before the Edinburg CISD school board for a vote this coming Tuesday.

  

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Ellis LuciaChristine Lagarde hands out hugs at St. Timothy Preschool in Mandeville, where she is employed.

Christine Lagarde loves her job. She likes passing out instruments in music class and helping the teachers hand out snacks. She likes teaching the hand motions that go along with the songs and comforting the toddlers.

If someone is sad, I sit with her and make her feel better, she says. What I love most about my job is the children.

Christine, 22, has Down syndrome. She started working at the St. Timothy United Methodist Church Preschool in October, and she thinks her job is perfect.

Her mother agrees.

Im just so happy shes happy, Kay Lagarde says. I couldnt have prayed for anything nicer.

Lynn Otillio, director of the Mandeville preschool, says Christines story shows how inclusion leads to success.

The Lagardes live in Covington now, but when Christine was a baby, they lived in Uptown New Orleans, where she was welcomed into a large, loving family. She has always gone to school with typical children.

She started out at Newcomb Nursery School and did really well there, Lagarde says. It was a very progressive school.

She and her husband, Jimmy, wanted Christine to continue in an inclusion program, and they asked administrators at Holy Name of Jesus School, where their sons were students, if Christine could go there.

She was the first child with Down syndrome at the school, and she was there from pre-k through fourth grade, Lagarde says. It was fabulous. She made her First Communion with her classmates and got so much out of it socially, but it was also great for the other kids.

When she was in second grade, Christine received the Cox Cable Inspirational Hero Award for befriending a little boy who was having all kinds of problems in school.

He kind of attached himself to her and did a turnaround, Lagarde says.

After the Lagardes moved to the north shore, Christine started going to public school in Mandeville and became an active member of Our Lady Queen of Peace Church. She was a cheerleader at Mandeville Junior High and joined the choir, and at Mandeville High School she was part of the Best Buddy Club, where typical students and special-needs teenagers hang out together and become friends.

Christines best buddy adores her, Lagarde says. Thats the beauty of inclusion. If you can get these kids together when theyre young, theyre accepting. Its just like being in a family.

Christine also got vocational training and life skills training during high school.

She went to a child development class she really liked, Lagarde says. She has always loved babies and children and older people. My dad was in skilled nursing over at Christwood, and she was nursing everyone in the place.

Before she graduated from Mandeville High in May, Christine was evaluated by Audrey Fabre, a vocational rehabilitation counselor, to see what kind of job would be best for her. And Fabre connected Christine with Lifeworks Career Development Center, which helps special-needs students find suitable jobs.

I wanted her to be somewhere like a retirement home or a nursery school, Lagarde says. Shes very nurturing. She just has a tender heart.

Otillio loves having Christine in the music room and the toddler room working with the children.

She just shines with her dear sweet spirit, the preschool director says.

Otillio knew Christine before she came to the school for an interview. A few years ago, when some mothers of special needs students were looking for a place to hold a summer camp for their kids, they approached St. Timothys and asked if they could hold it there. Otillios daughter ended up running Camp Tiger Paw, and Otillio helped out.

We just feel so blessed that Lynn got to know Christine through the little camp, Lagarde says.

When Otillio presented the idea of hiring Christine to the preschool board, members were all for it.

She had the interview on Friday, and they let her start on Monday, Lagarde says. I think this is God telling me, This is what Christine was meant to do.

For Otillio, who has worked at St. Timothys for 25 years, having Christine on her staff is a natural progression. Her first experience with inclusion came in the early 90s when Stephanie Frazer came to her and asked if her son Christopher, who had physical disabilities caused by a rare neurological disorder, could be enrolled in the school if he had a personal attendant to help with his needs.

Frazer had been driving her son across the lake to be in a state-of-the-art inclusive program at LSU Medical School where he was in a class of five typical and five special-needs students. The program was for children up to age 3, and she wanted him to continue in an inclusive program where typical children would serve as role models.

Otillio remembers the day they came to meet her and see the school.

Christopher was using a little walker reluctantly, but when we went into the gym and he saw the other children, he grabbed the sides of it and took off, she says.

He started school the following September and blossomed there.

Otillios most vivid memory of Christopher is at the annual Christmas program three months later when all the children filed up to the front of the church to sing.

There was Christopher, walking independently with his classmates, she says. When you see the children in their little white Christmas collars, its always your special Christmas moment. That year, I dont think there was a dry eye in the sanctuary.

Frazer says Christophers experience at St. Timothys was very positive for him and comforting for her.

They were able to look beyond his differences and integrate him into their activities, she says. He started out with that little walker and ended up carrying it around.

Christopher is now a student at Mandeville High School.

I dont think he would have progressed like he has without the opportunity at St. Timothys, Frazer says. It was good for us, and it was good for them.

Several other special-needs students have attended St. Timothys since Christopher went there, some with personal attendants and some who didnt need them.

Every child has been different, and their needs have been different, Otillio says. We dont offer therapy or special education. We offer exposure to typical kids.

Whats overlooked sometimes, she says, is the wonderful effect inclusion has on typical children.

They learn not to be afraid of someone different, she says. Our September unit is always about how God made each one of us special and unique. The special-needs children have made a huge difference in all of us.

And now Christine is at St. Timothys, filling the school with her dear sweet spirit.

I never dreamed we would have a special needs child, and now we employ a special-needs adult, Otillio says, smiling. I feel like weve come full-circle. It just doesnt get any better than that.

Sheila Stroup’s column appears Sunday, Tuesday and Thursday in Living. Contact her at or 985.898.4831.

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We are excited to let you know that PTO Today and The Savings Bank Life Insurance Company of Massachusetts (SBLI) have teamed up to celebrate school and parent group volunteers from across the country. You may feel sometimes that your tireless volunteer efforts go unnoticed. Take heart, PTO Today and SBLI think that volunteers deserve recognition beyond their school walls. Starting this month, we’re kicking off a joint program called Superstar Volunteers. As part of this program, we profile a new Superstar Volunteer and their unique volunteer story every two weeks, from January through September. Here’s the really cool part: SBLI will donate $500 to the school of each volunteer we profile.

Our first Superstar Volunteer is Sunnie Robles-Schmidt from Santa Rosa, California. Props to Sunnie who has figured out how to juggle her busy family life AND volunteer at both the elementary school and high school. to learn more about what kind of volunteer work brings her joy. Congrats Sunnie, for being the first SBLI Superstar Volunteer and for winning $500 for your school!

Do you have a volunteer at your school that you think deserves a SBLI Superstar Volunteer designation? today – winners get a $500 donation to their school and will be featured on both ptotoday.com and SBLI, just like Sunnie!

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