3.2-4.1 Geometry Notes and Review PDF

3.2 Notes: Proving lines are parallel Learning target: use theorems to prove two lines are parallel  Learning target: use theorems to prove two lines are parallel Through a point not on a line, there is one and only one line parallel to the given...

3.2 Notes: Proving lines are parallel Learning target: use theorems to prove two lines are parallel  Learning target: use theorems to prove two lines are parallel Through a point not on a line, there is one and only one line parallel to the given line. There is exactly one line through P parallel to l.  Learning target: use theorems to prove two lines are parallel Through a point not on a line, there is one and only one line perpendicular to the given line.  Learning target: use theorems to prove two lines are parallel Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. If… Then… m ⊥ t and n ⊥ t m!n  Learning target: use theorems to prove two lines are parallel Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. If… Then… ∠4 ≅ ∠6 ℓ"m  Learning target: use theorems to prove two lines are parallel Which lines are parallel if ∠1 ≅ ∠7? Justify your answer.  Learning target: use theorems to prove two lines are parallel Theorem If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. If… Then… ∠2 ≅ ∠6 ℓ"m  Learning target: use theorems to prove two lines are parallel Theorem If two lines and a transversal form same- side interior angles that are supplementary, then the two lines are parallel. If… Then… ! ℓ"m m∠3 + m∠6 = 180  Learning target: use theorems to prove two lines are parallel Theorem If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. If… Then… ∠1 ≅ ∠7 ℓ"m Copyright © 2014 Pearson Education, Inc. Learning target: use theorems to prove two lines are parallel Thus far, we have learned two forms of proof— paragraph and two-column (statement/ reason). In a third form, called flow proof, arrows show the logical connections between the statements. Reasons are written below the statements. Here we show the same proof as above, but in flow-proof form.  Learning target: use theorems to prove two lines are parallel Given: ∠1 ≅ ∠7 Prove: ℓ " m What We Know: ∠1 ≅ ∠7 What We Need: ∠1 and ∠3 are vertical One pair of ∠5 and ∠7 are vertical corresponding or ∠1 and ∠5 are corresponding alternate interior angles congruent to ∠3 and ∠7 are corresponding prove l || m ∠3 and ∠5 are alternate interior  Learning target: use theorems to prove two lines are parallel Given: ∠1 ≅ ∠7 Prove: ℓ " m What to Do: Use a pair of congruent vertical angles to relate either angle 1 or angle 7 to its corresponding angle. Learning target: use theorems to prove two lines are parallel Given: ∠1 ≅ ∠7 Prove: ℓ " m Flow Proof: ∠1 ≅ ∠7 ∠3 ≅ ∠7 ℓ"m Given Substitution (or If corresponding ∠3 ≅ ∠1 angles are Transitive Vertical angles Property) congruent, then are congruent. the lines are parallel.  Learning target: use theorems to prove two lines are parallel The fence gate at the right is made up of pieces of wood arranged in various directions. Suppose ∠1 ≅ ∠2. Are lines r and s parallel? Explain.  Learning target: use theorems to prove two lines are parallel What is the value of x that makes a || b?  3.3 Notes: Parallel lines and angle formed by transversals Learning Target: Prove and use theorems about parallel lines cut by transversals  Learning Target: Prove and use theorems about parallel lines cut by transversals Theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If… Then… ∠4 ≅ ∠6 ℓ"m ∠3 ≅ ∠5  Learning Target: Prove and use theorems about parallel lines cut by transversals Using the figure shown and given that m∠3 = 55! , find the measure of each angle. Tell what theorem or postulate you used. (Recall that red arrow head notation means || lines.) a. m∠5 b. m∠7 c. m∠4 d. m∠2  Learning Target: Prove and use theorems about parallel lines cut by transversals Theorem If two parallel lines are cut by a transversal, then corresponding angles are congruent. If… Then… ∠1 ≅ ∠5 ℓ"m ∠2 ≅ ∠6 ∠3 ≅ ∠7 ∠4 ≅ ∠8  Learning Target: Prove and use theorems about parallel lines cut by transversals Theorem If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. If… Then… ! ℓ"m m∠4 + m∠5 = 180 ! m∠3 + m∠6 = 180  Learning Target: Prove and use theorems about parallel lines cut by transversals Theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent. If… Then… ∠1 ≅ ∠7 ℓ"m ∠2 ≅ ∠8  Learning Target: Prove and use theorems about parallel lines cut by transversals ! Given the figure shown and m∠4 = 42 , find the measures of the other angles.  Learning Target: Prove and use theorems about parallel lines cut by transversals Using the figure shown, what are the measures of angle 3 and angle 4? Which theorem justifies each answer? Learning Target: Prove and use theorems about parallel lines cut by transversals Given the figure and l || m, find the value of x.  4.1 Notes: Types of Triangles Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles A triangle is formed by The noncollinear points are called The segments joining the points are called Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles We call this triangle:_________ Sides:_____________________ Vertices:____________________  Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Given ΔPQR: a. Which angle is opposite segment PQ?__________ b. Which side is opposite angle Q?____________ c. Which side is included between angle P and angle R?__________ d. Which angle is included between segments QR and PR? ___________  Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Classification by Angles Type Description Example Acute Obtuse Equiangular Right Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Classification by Sides Type Description Example Scalene Isosceles Equilateral Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Classify each triangle by its angles and sides. Use the most specific name.  Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Notes:_____________ ___________________ ___________________ ___________________ ___________________ ___________________ ___________________ Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Triangle Angle-Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. m∠1 + m∠2 + m∠3 = 180°  Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Exterior Angle of a Triangle Notes:_____________ ___________________ ___________________ ___________________ ___________________ ___________________ ___________________ Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Acute Angles of a Right Triangle Notes:_____________ ___________________ ___________________ ___________________ ___________________ ___________________ ___________________  Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Use the Triangle Angle-Sum Theorem to find the measure of each angle in the given triangle. Notes:_____________ ___________________ ___________________ ___________________ ___________________ ___________________ ___________________  Learning Target: Learn the Vocabulary of Triangles, Classify triangles by angles and sides,Find measures of triangles Use the Exterior Angle of a Triangle to find the measure of the exterior angle and the nonadjacent angle shown. Notes:_____________ ___________________ ___________________ ___________________ ___________________ ___________________ ___________________  Angle Pairs Created by Parallel Lines Cut by a Transversal Vocabulary  ____________________ - A line that crosses parallel lines to create pairs of congruent and supplementary angles  ____________________ - Having the same measurement  ____________________ - Angles that add up to 180° Angle Pairs in Parallel Lines Cut by a Transversal Exterior Exterior Interior Interior Exterior Exterior  ____________________ - Angles that lie on the same side of the transversal and on the same side of the parallel lines. These angles are in the same “corner” and are congruent  ____________________ - Angles on opposite sides of the transversal and inside the two parallel lines. These angles are congruent  ____________________ - Angles on opposite sides of the transversal and outside the parallel lines. These angles are congruent  ____________________ - Angles on the same side of the transversal and inside the parallel lines. These angles are supplementary  ____________________ - Angles on the same side of the transversal and outside the parallel lines. These angles are supplementary  ____________________ - Angles that are across from each other and are formed by any intersecting lines (not just parallel lines and transversals). These angles are congruent. ©Amazing Mathematics 3  Angle Pairs Created by Parallel Lines Cut by a Transversal Cut out the squares on page 5 and glue them to the corresponding square Fill in “Congruent” or “Supplementary” for each box to say what the angles are Glue Flap Here Glue Flap Here Glue Flap Here Corresponding Angles Same-Side Exterior Angles Vertical Angles The angles are The angles are The angles are _________________ _________________ _________________ Glue Flap Here Glue Flap Here Glue Flap Here Same-Side Interior Angles Alternate Interior Angles Same-Side Interior Angles The angles are The angles are The angles are _________________ _________________ _________________ Glue Flap Here Glue Flap Here Glue Flap Here Alternate Exterior Angles Vertical Angles Alternate Exterior Angles The angles are The angles are The angles are _________________ _________________ _________________ Glue Flap Here Glue Flap Here Glue Flap Here Alternate Interior Angles Same-Side Exterior Angles Corresponding Angles The angles are The angles are The angles are _________________ _________________ _________________ ©Amazing Mathematics 4  Angle Pairs Created by Parallel Lines Cut by a Transversal 1. Cut out the squares Do NOT cut the dotted 2. Fold along the dotted line in each square to create a “flap” lines.. they are for folding 3. Glue the flap to the corresponding square on page 4 ©Amazing Mathematics 5  Angle Pairs Created by Parallel Lines Cut by a Transversal For each set of angles name the angle pair and find the missing measurement 1) Type of angle pair 5) Type of angle pair ______________ ______________ 68° These angles are x° These angles are ______________ ______________ x° so….x=________ 77° so….x=________ 2) Type of angle pair 6) Type of angle pair ______________ ______________ These angles are 106° These angles are x° ______________ ______________ x° 134° so….x=________ so….x=________ 3) Type of angle pair 7) Type of angle pair 120° ______________ ______________ x° These angles are 74° These angles are ______________ x° ______________ so….x=________ so….x=________ 4) Type of angle pair 8) Type of angle pair ______________ ______________ 101° These angles are x° These angles are ______________ 142° ______________ x° so….x=________ so….x=________ ©Amazing Mathematics 6  Angle Pairs Created by Parallel Lines Cut by a Transversal For each set of angles name the angle pair, write the equation, solve the equation for x, and plug in x to find the missing angle measurements 1) Type of angle pair _____________ Show your work These angles are ______________ 3x° Equation ____________________ 6x° x=________ Angle Measurements= ___________ 2) Show your work Type of angle pair _____________ 7x-12° These angles are ______________ Equation ____________________ 3x+28° x=________ Angle Measurements= ___________ 3) Type of angle pair _____________ Show your work 3x+77° These angles are ______________ Equation ____________________ 4x+54° x=________ Angle Measurements= ___________ 4) Type of angle pair _____________ Show your work These angles are ______________ 9x+8° Equation ____________________ x=________ 4x+18° Angle Measurements= ___________ ©Amazing Mathematics 7  Kuta Software - Infinite Geometry Name___________________________________ Classifying Triangles Date________________ Period____ Classify each triangle by each angles and sides. Base your decision on the actual lengths of the sides and the measures of the angles. 1) 2) 3) 4) 5) 6) ©k s200C1B1Y pKMuptUa4 ASRoBfptqwGaDrMea 4LjLGCd.e F kAvlulL Nr1iMgwhZtesa frZe8sOeyrAvMeld0.C v 7MUabdaer gwNigtlhu 2I1n7feiEntiMt1e0 CGkedoIm3evtdr8yP.v -1- Worksheet by Kuta Software LLC  Classify each triangle by each angles and sides. 7) 8) 60° 57° 8.6 8.6 8.7 6.1 60° 60° 79° 44° 8.6 7.4 9) 10) 26° 4.5 32° 13.2 2.5 11.2 128° 26° 2.5 90° 58° 7 11) 12) 4.8 72° 36° 4.8 4.8 3 45° 45° 72° 4.8 6.8 Classify each triangle by each angles and sides. Equal sides and equal angles, if any, are indicated in each diagram. 13) 14) ©G r2q0m1o1a pKeuytXa8 CSLoJf7tDwZasrHeC GL3LqCz.b r DAClAla trHiJgahctZsq irjekseehrnvHe4dJ.r 2 7MzafdFeY Nwki9tVhV KIlnGfkiSn6iMtIer FGDekoamTejtHr3yS.l -2- Worksheet by Kuta Software LLC  15) 16) 17) 18) Sketch an example of the type of triangle described. Mark the triangle to indicate what information is known. If no triangle can be drawn, write "not possible." 19) acute isosceles 20) right scalene 21) right isosceles 22) right equilateral 23) acute scalene 24) obtuse scalene 25) right obtuse 26) equilateral ©0 92D0d1i1S pKau5tiab ISSoifutfwva4rcer MLuLGCp.K c tAoldlY UraiigGhatVsS qrQeLs1eKrCvoe1dp.U M eM0aldje8 gwOiTtehu qIunIfxihntiwtpe3 mGvepo7myeotXrdyj.X -3- Worksheet by Kuta Software LLC  Chapter 3 Parallel and Perpendicular Lines Study Guide 3.1 Identify Pairs of Lines/Angles 3.2- Parallel Lines and Transversals Parallel Lines **Know which angles are congruent Parallel Postulate and supplementary Perpendicular Postulate Skew Lines Corresponding Angles Postulate Parallel Planes Alternate Interior Angles Theorem Diagram with a cube/box Alternate Exterior Angles Theorem Transversals Consecutive Interior- (Same Side Angles formed by transversals Interior) Angles Theorem Corresponding Angles Alternate Interior Angles **Know more difficult problems Alternate Exterior Angles with multiple lines, systems of Consecutive Interior- (Same Side equations and factoring! (we had 2 Interior) Angles worksheets on this!) 3.3 Proving Lines Parallel 3.6 Perpendicular Lines **Converses used to show lines are Theorem 3.8- Two lines intersect to form PARALLEL a linear pair of congruent angles, then the lines are perpendicular Corresponding Angles Converse Alternate Interior Angles Converse Theorem 3.9- If 2 lines are Alternate Exterior Angles Converse perpendicular, then they intersect to Consecutive Interior- (Same Side form 4 right angles Interior) Angles Converse Transitive Property of Parallel Lines Right Angle Pair Theorem (3.10)- Two angles that make a right angle pair are complementary **Don’t Forget About: Perpendicular Transversal Theorem- If a Linear Pairs- Supplementary transversal is perpendicular to one of Vertical Angles- Congruent two parallel lines, then it is perpendicular to the other Lines Perpendicular to a Transversal Theorem- If two lines are perpendicular to the same line, then they are perpendicular to each other Part I: Circle the word that best completes the sentence. 1. If two lines are parallel, then they (ALWAYS…..SOMETIMES…..NEVER) intersect. 2. If one line is skew to another, then they are (ALWAYS…..SOMETIMES…..NEVER) coplanar. 3. If two lines intersect, then they are (ALWAYS…..SOMETIMES…..NEVER) perpendicular. 4. If two lines are coplanar, then they are (ALWAYS…..SOMETIMES…..NEVER) parallel. 5. If two lines are cut by a transversal such that the alternate interior angles are (CONGRUENT…..COMPLEMENTARY…..SUPPLEMENTARY), then the lines are parallel. 6. If two lines are cut by a transversal such that the consecutive interior angles are (CONGRUENT…..COMPLEMENTARY…..SUPPLEMENTARY), then the lines are parallel. 7. If two lines are cut by a transversal such that the corresponding angles are (CONGRUENT…..COMPLEMENTARY…..SUPPLEMENTARY), then the lines are parallel. Part II: Think of each segment in the diagram as part of a line. Complete the statement with PARALLEL, SKEW, or PERPENDICULAR. 1. ⃡ ⃡ are _______________________ 2. ⃡ ⃡ are _______________________ 3. ⃡ ⃡ are _______________________ 4. and are _______________________ 5. and are _______________________ Part III: Classify the angle pair as corresponding angles, alternate interior angles, alternate exterior angles, same side (consecutive) interior angles, vertical angles, linear pair, or none. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Part IV: Find the value of the variables. 1. 2. 3. ( ( 4. 5. Part V. Is there enough information to state that lines and are parallel? If so, state the reason. 1. Yes_________ No__________ Reason (if necessary)____________________________ _______________________________________________ 2. Yes_________ No__________ Reason (if necessary)____________________________ ______________________________________________ 3. Yes_________ No__________ Reason (if necessary)____________________________ _______________________________________________ Part VI. Use the diagram and the given information to determine if , or neither. 1. _______________ 2. _______________ 3. _______________ 4. _______________ 5. _______________ 6. _______________ 7. _______________ Part VII. Find the measure of the indicated angle. 12 6 5 4 3 1. _________________ 2. _______________________ 3. __________________ 4. _______________________ 5. __________________ 6. _______________________ Part VIII. Use the diagram. 1. Is r ? Yes__________ No___________ 2. Is Yes__________ No___________ 3. Is r Yes__________ No___________ Part IX. In the diagram, ⃡ ⃡. Find the value of. 1. R S T ____________ 2. R S T ____________ 3. R S T ____________  Kuta Software - Infinite Geometry Name___________________________________ Parallel Lines and Transversals Date________________ Period____ Identify each pair of angles as corresponding, alternate interior, alternate exterior, or consecutive interior. 1) 2) y y x x 3) 4) y y x x 5) 6) y y x x 7) 8) y x x y ©e Z2t0h1Z1k qKUuNtraS ZSZoHfRtqwnacr6e5 eLKLSCZ.C U 3ASl1lL Qr3iRguhNt2sE srIeYs0eIrXvYePd2.7 Z xMkakdJeO lwaiItWh9 tIxnvf9iCnxiGtnes LGKecoTmTeZthrxyG.0 -1- Worksheet by Kuta Software LLC  9) 10) y x x y Find the measure of each angle indicated. 11) 12) ? 84° ? 110° 13) 14) 100° ? ? 111° 15) 16) ? 125° ? 47° 17) 18) 53° ? 113° ? ©i L2c0j1e12 wKcuXtwaN NSmoFfftDwxaTr1eS OLwLXCZ.4 b GAKlzlx 7rHi1gVhntisE 5r1eJsge5rwvteOd3.o Z TMTaOd0eF DwQihtohB TInnAfViYnRi5tveR 4G1eTo7mUeitOrGyL.k -2- Worksheet by Kuta Software LLC  Solve for x. 19) 20) 75° 11 x − 2 21 x + 6 21) 22) 60° x + 139 8x − 4 132° 23) 24) −1 + 14 x 23 x − 5 12 x + 17 21 x + 5 Find the measure of the angle indicated in bold. 25) 26) x + 96 20 x + 5 x + 96 24 x − 1 27) 28) 6x x + 109 x + 89 5 x + 10 ©F z2O0A1H1N EKTuttpaK YS1o3fQt4wIahryeN gL1LFCl.s W tA1lql4 krriwgbhztjsh krNe6smeVrevVeBdI.1 Q yMNa9djeL ew4iPtPhT kINnAf2iQnjiQtmeq eGjeiopmye8tVrJyM.o -3- Worksheet by Kuta Software LLC