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High School Geometry: Tutoring Solution14 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Mia Primas*

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

When someone claims something is true, it's important that they are able to support the claim with proof. In this lesson, you will learn how flow proofs are used in geometry to support a mathematical claim.

Throughout the history of mathematics, a **proof** has been a series of statements that lead to a conclusion. Proofs begin with one or more given statements, which are provided. The given statement leads to other statements until the desired conclusion is reached. Each statement in a sequence must be supported with logical reasoning. Mathematical properties, definitions, and theorems are used to validate the statements.

When working with geometry proofs, it is important to be familiar with properties, definitions, and theorems that can be used to validate each statement. For example, if you're asked to prove that two triangles are congruent, it's useful to know the theorems of congruent triangles. Or, if a proof involves a rectangle, you may need to use the properties of the sides and angles of rectangles to support your reasoning.

Proofs can be presented in different formats, such as a paragraph, a two-column chart, or a flow chart. In a **paragraph proof**, the statements and reasons are written as sentences. In a **two-column proof**, the statements are written in one column, and the reasons are written next to them in a second column. A **flow proof** uses a diagram to show each statement leading to the conclusion. Arrows are drawn to represent the sequence of the proof. The layout of the diagram is not important, but the arrows should clearly show how one statement leads to the next. The explanation for each statement is written below the statement.

The following examples will detail each proof in paragraph form, followed by a flow chart. The examples begin with the given information, along with the desired conclusion to be proven.

Our first example begins with the figure of quadrilateral *ABCD* and the given statement telling us that it's a rectangle. One of the properties of rectangles is that the opposite sides are congruent to each other. We can use this to show that side *AD* is congruent to side *BC* and side *AB* is congruent to side *CD*. The reflexive property tells us that any line segment is congruent to itself; therefore, side *AC* is congruent to side *AC*. We now have three pairs of congruent sides. Using the Side-Side-Side theorem of triangle congruency, we've proven that triangle *ACD* is congruent to triangle *CAB*.

We are told that line *FH* is the **perpendicular bisector** of line *EG*. From this statement, we know that line *FH* is perpendicular to line 'EG', forming two right angles (angles *FHE* and *FHG*). Because all right angles are congruent, angles *FHE* and *FHG* are congruent to each other. We also know from the given statement that line *FH* **bisects** line *EG*, meaning that it cuts line *EG* into two congruent parts (lines *EH* and *HG*). So far, we have a pair of congruent angles and a pair of congruent lines. We can use the reflexive property to show that line *FH* is congruent to line *FH*. This is enough information to prove that triangle *EFH* is congruent to triangle *GFH* according to the Side-Angle-Side theorem of congruent triangles. This means that each part of triangle *EFH* is congruent with its corresponding part in triangle *GFH*. From this, we can conclude that side *EF* is congruent to side *FG*, making triangle *EFG* an isosceles triangle.

Mathematical **proofs**, or a series of statements that lead to a conclusion, use logical reasoning to support that a claim is true. When writing geometric proofs, it's important to have a good understanding of the definitions (like **bisection**, which is when one line cuts another line into two congruent parts), properties, and theorems used in geometry. A **flow proof** is one of the ways that a proof can be formatted. The other two are a **paragraph proof** (in which the statements and reasons are written as sentences) and a **two-column proof** (in which the statements are written in one column and the reasons are written next to them in a second column). A flow proof, though, uses a diagram with arrows showing each statement leading to the desired conclusion.

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High School Geometry: Tutoring Solution14 chapters | 160 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:16
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Angle of Elevation: Definition, Formula & Examples 4:50
- Centroid: Definition, Theorem & Formula 6:15
- Flow Proof in Geometry: Definition & Examples 4:31
- Law of Detachment in Geometry: Definition & Examples 4:30
- Phase Shift: Definition & Formula
- Side-Side-Side (SSS) Triangle: Formula & Theorem
- Square Pyramid: Definition & Properties 3:59
- What is an Acute Triangle? - Definition, Facts & Example 2:44
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